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Meromorphic function on the complex plane
An L-function is a meromorphic function on the complex plane, and one out of several categories of mathematical objects studied in analytic number theory
L-function
Mathematical concept
mathematics, the explicit formulae for L-functions are relations between sums over the complex number zeroes of an L-function and sums over prime powers, introduced
Explicit formulae for L-functions
Explicit_formulae_for_L-functions
Type of mathematical function
In mathematics, a Dirichlet L-series is a function of the form L ( s , χ ) = ∑ n = 1 ∞ χ ( n ) n s , {\displaystyle L(s,\chi )=\sum _{n=1}^{\infty }{\frac
Dirichlet_L-function
Topics referred to by the same term
In mathematics, a Hecke L-function may refer to: an L-function of a modular form an L-function of a Hecke character This disambiguation page lists mathematics
Hecke_L-function
Type of Dirichlet series associated to number field extensions
In mathematics, Artin L-functions are a type of Dirichlet series defined for finite extensions of number fields, encoding informations about linear representations
Artin_L-function
Mathematical concept
In mathematics, an automorphic L-function is a function L(s,π,r) of a complex variable s, associated to an automorphic representation π of a reductive
Automorphic_L-function
Analytic function in mathematics
Riemann zeta function, such as Dirichlet series, Dirichlet L-functions and L-functions, are known. The Riemann zeta function ζ(s) is a function of a complex
Riemann_zeta_function
mathematics, motivic L-functions are a generalization of Hasse–Weil L-functions to general motives over global fields. The local L-factor at a finite place
Motivic_L-function
Space of bounded sequences
Σ , μ ) {\displaystyle L^{\infty }=L^{\infty }(X,\Sigma ,\mu )} , the vector space of essentially bounded measurable functions with the essential supremum
L-infinity
Method of solution to differential equations
that if L {\displaystyle L} is a linear differential operator, then the Green's function G {\displaystyle G} is the solution of the equation L G = δ ,
Green's_function
Mathematical concept
In mathematics, the term standard L-function refers to a particular type of automorphic L-function described by Robert P. Langlands. Here, standard refers
Standard_L-function
p-adic zeta function, or more generally a p-adic L-function, is a function analogous to the Riemann zeta function, or more general L-functions, but whose
P-adic_L-function
Artin L-function is a function associated to a finite Galois extension of global fields created by packaging together the various Artin L-functions associated
Equivariant_L-function
S-shaped curve
A logistic function or logistic curve is a common S-shaped curve (sigmoid curve) with the equation f ( x ) = L 1 + e − k ( x − x 0 ) {\displaystyle f(x)={\frac
Logistic_function
Mathematical function associated to algebraic varieties
global L-function defined as an Euler product of local zeta functions. Hasse–Weil L-functions form one of the two major classes of global L-functions, alongside
Hasse–Weil_zeta_function
Conjecture on zeros of the zeta function
Unsolved problem in mathematics Do all non-trivial zeros of the Riemann zeta function have a real part equal to one half? More unsolved problems in mathematics
Riemann_hypothesis
Subfield of number theory
In mathematics, the study of special values of L-functions is a subfield of number theory devoted to generalising formulae such as the Leibniz formula
Special_values_of_L-functions
In mathematics, the Shimizu L-function, introduced by Hideo Shimizu in 1963, is a Dirichlet series associated to a totally real algebraic number field
Shimizu_L-function
Function studied by Ramanujan
of weight 12, it gives rise to an L {\displaystyle L} -function, called Ramanujan's L {\displaystyle L} -function. It is defined for R e ( s ) > 13 /
Ramanujan_tau_function
Function whose squared absolute value has finite integral
square-integrable function, also called a quadratically integrable function or L 2 {\displaystyle L^{2}} function or square-summable function, is a real- or
Square-integrable_function
function Ihara zeta function of a graph L-function, a "twisted" zeta function Lefschetz zeta function of a morphism Lerch zeta function, a generalization
List_of_zeta_functions
Point to which functions converge in analysis
f(x) to every input x. We say that the function has a limit L at an input p, if f(x) gets closer and closer to L as x moves closer and closer to p. More
Limit_of_a_function
Unsolved problem in mathematics
plane, the Ramanujan L-function can be defined by analytic continuation of this series. Like other L-functions, the Ramanujan L-function satisfies a functional
Ramanujan–Petersson conjecture
Ramanujan–Petersson_conjecture
Mathematical function whose derivative exists
a real function f {\displaystyle f} , then f {\displaystyle f} is said to be differentiable at x 0 {\displaystyle x_{0}} if there exists an L ∈ R {\displaystyle
Differentiable_function
Special mathematical function
is a particular Dirichlet L-function, the L-function for the alternating character of period four. The Dirichlet beta function is defined as β ( s ) = ∑
Dirichlet_beta_function
In mathematics, a Shintani zeta function or Shintani L-function is a generalization of the Riemann zeta function. They were first studied by Takuro Shintani (1976)
Shintani_zeta_function
Mathematical function used in signal processing
processing, the function is sampled symmetrically (with spacing L / N {\displaystyle L/N} and amplitude 1 {\displaystyle 1} ): w [ n ] = L ⋅ w 0 ( L N ( n − N
Hann_function
Generalization of the Riemann zeta function for algebraic number fields
L / K ) {\displaystyle {\text{Gal}}(L/K)} , the resulting Artin L-function is: L ( s , 1 , L / K ) = ζ K ( s ) . {\displaystyle L(s,{\mathcal {1}},L/K)=\zeta
Dedekind_zeta_function
Algebraic curve in mathematics
function of a complex variable, L, the Hasse–Weil zeta function of E over Q. This function is a variant of the Riemann zeta function and Dirichlet L-functions
Elliptic_curve
Conjectures connecting number theory and geometry
L-function. One of his conjectures states that these L-functions satisfy a certain functional equation generalizing those of other known L-functions.
Langlands_program
Mathematical relation assigning a probability event to a cost
optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one
Loss_function
Function used in signal processing
processing and statistics, a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside
Window_function
Type of character in number theory
to construct a class of L-functions larger than Dirichlet L-functions, and a natural setting for the Dedekind zeta-functions and certain others which
Hecke_character
In mathematics, the L-functions of number theory are expected to have several characteristic properties, one of which is that they satisfy certain functional
Functional equation (L-function)
Functional_equation_(L-function)
Mathematical conjecture about zeros of L-functions
zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global L-functions, which are formally similar
Generalized Riemann hypothesis
Generalized_Riemann_hypothesis
Transcendental single-variable function
tangent integral, polygamma function, Riemann zeta function, Dirichlet eta function, and Dirichlet beta function. The Clausen function of order 2 – often referred
Clausen_function
Mathematical description of quantum state
In quantum mechanics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common
Wave_function
representations of L-functions, is a technique for directly constructing and analytically continuing several important examples of automorphic L-functions. Some authors
Rankin–Selberg_method
Mathematical conjecture
Özlük, proved the pair correlation conjecture for some of Dirichlet's L-functions (A. E. Ozluk (1982)). The connection with random unitary matrices could
Montgomery's pair correlation conjecture
Montgomery's_pair_correlation_conjecture
Unproved conjecture in mathematics
{\displaystyle K} and the behaviour of its associated Hasse–Weil L-function L ( E , s ) {\displaystyle L(E,s)} at s = 1 {\displaystyle s=1} . More specifically
Birch and Swinnerton-Dyer conjecture
Birch_and_Swinnerton-Dyer_conjecture
Function specifying the behavior of a component in an electronic or control system
a transfer function (also known as system function or network function) of a system, sub-system, or component is a mathematical function that models
Transfer_function
Special function in mathematics
In mathematics, the Hurwitz zeta function is one of the many zeta functions. It is formally defined for complex variables s with Re(s) > 1 and a ≠ 0, −1
Hurwitz_zeta_function
Ratio of polynomial functions
is L. The set of rational functions over a field K is a field, the field of fractions of the ring of the polynomial functions over K. A function f {\displaystyle
Rational_function
Mathematic theory
locally compact group of ideles to lift the zeta function twisted by a Hecke character, i.e. a Hecke L-function, of a number field to a zeta integral and study
Tate's_thesis
Sigmoid shape special function
mathematics, the error function (also called the Gauss error function), often denoted by e r f {\displaystyle \mathbf {erf} } , is the function erf ( z ) = 2
Error_function
The Selberg zeta-function was introduced by Atle Selberg (1956). It is analogous to the famous Riemann zeta function ζ ( s ) = ∏ p ∈ P 1 1 − p − s {\displaystyle
Selberg_zeta_function
Function related to statistics and probability theory
density function f {\textstyle f} (a function of x {\textstyle x} ) which depends on a parameter θ {\textstyle \theta } . Then the function L ( θ ∣ x
Likelihood_function
Mathematical function
In mathematics, the prime zeta function is an analogue of the Riemann zeta function, studied by Glaisher (1891). It is defined as the following infinite
Prime_zeta_function
Special case of the polylogarithm
Spence's function), denoted as Li2(z), is a particular case of the polylogarithm. Two related special functions are referred to as Spence's function, the
Dilogarithm
Association of one output to each input
mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the
Function_(mathematics)
Methods used in statistics
variance stabilized Ripley K function called the L function is generally used. The sample version of the L function is defined as L ^ ( t ) = ( K ^ ( t ) π
Spatial descriptive statistics
Spatial_descriptive_statistics
Multivalued function in mathematics
) 6 L 1 3 + L 2 ( − 12 + 36 L 2 − 22 L 2 2 + 3 L 2 3 ) 12 L 1 4 + ⋯ = L 1 − L 2 + ∑ ℓ = 1 ∞ 1 L 1 ℓ ∑ m = 1 ℓ ( − 1 ) ℓ − m [ ℓ ℓ − m + 1 ] m ! L 2 m
Lambert_W_function
Extension of the factorial function
The gamma function then is defined in the complex plane as the analytic continuation of this integral function: it is a meromorphic function which is holomorphic
Gamma_function
Theorem on the number of primes in arithmetic sequences
Dirichlet (1837) with Dirichlet L-series. The proof is modeled on Euler's earlier work relating the Riemann zeta function to the distribution of primes
Dirichlet's theorem on arithmetic progressions
Dirichlet's_theorem_on_arithmetic_progressions
Mathematical function having a characteristic S-shaped curve or sigmoid curve
sigmoid function is any mathematical function whose graph has a characteristic S-shaped or sigmoid curve. A common example of a sigmoid function is the
Sigmoid_function
Summability method in physics
In mathematics and theoretical physics, zeta function regularization is a type of regularization or summability method that assigns finite values to divergent
Zeta_function_regularization
Arithmetic function related to the divisors of an integer
theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as the divisor function, it counts the number
Divisor_function
Function that is discontinuous at rationals and continuous at irrationals
Thomae's function is a real-valued function of a real variable that can be defined as: f ( x ) = { 1 q if x = p q ( x is rational), with p ∈ Z and
Thomae's_function
Sum of inverse squares of natural numbers
orthonormal basis in the space L per 2 ( 0 , 1 ) {\displaystyle L_{\operatorname {per} }^{2}(0,1)} of L2 periodic functions over ( 0 , 1 ) {\displaystyle
Basel_problem
Mathematical series
definition of the Riemann zeta function is a Dirichlet series, as are the Dirichlet L-functions. Specifically, the Riemann zeta function ζ(s) is the Dirichlet
Dirichlet_series
Zeta-like functions approximate arbitrary holomorphic functions
universality of zeta functions is the remarkable ability of the Riemann zeta function and other similar functions (such as the Dirichlet L-functions) to approximate
Zeta_function_universality
Program function without side effects
In computer programming, a pure function is a function that has the following properties: the function return values are identical for identical arguments
Pure_function
Generalized function whose value is zero everywhere except at zero
Dirac delta function (or δ {\displaystyle {\boldsymbol {\delta }}} distribution), also known as the unit impulse, is a generalized function on the real
Dirac_delta_function
elliptic curve. The question of the rank is thought to be bound up with L-functions (see below). The torsor theory here leads to the Selmer group and Tate–Shafarevich
Arithmetic of abelian varieties
Arithmetic_of_abelian_varieties
mathematics, the local zeta function Z(V, s) (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as Z ( V , s ) =
Local_zeta_function
Complex-valued arithmetic function
theory and related branches of mathematics, a complex-valued arithmetic function χ : Z → C {\displaystyle \chi :\mathbb {Z} \rightarrow \mathbb {C} } is
Dirichlet_character
Topics referred to by the same term
meromorphic function on the complex plane L {\displaystyle {\mathcal {L}}} , Laplace transform L {\displaystyle {\mathcal {L}}} , likelihood function ℓp space
L_(disambiguation)
In mathematics, the Brownian motion and the Riemann zeta function are two central objects of study in mathematics originating from different fields - probability
Brownian motion and Riemann zeta function
Brownian_motion_and_Riemann_zeta_function
Special mathematical functions defined on the surface of a sphere
[ L z , L + ] = L + , [ L z , L − ] = − L − , [ L + , L − ] = 2 L z . {\displaystyle [L_{z},L_{+}]=L_{+},\quad [L_{z},L_{-}]=-L_{-},\quad [L_{+},L_{-}]=2L_{z}
Spherical_harmonics
Mathematical function
In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form f ( x ) = exp ( − x 2 ) {\displaystyle f(x)=\exp(-x^{2})}
Gaussian_function
Ratio of the perimeter of Bernoulli's lemniscate to its diameter
}{\frac {\nu (n)}{n}}={\frac {\varpi }{4}}} where L {\displaystyle L} is the L-function of the elliptic curve E : y 2 = x 3 − x {\displaystyle E:\,y^{2}=x^{3}-x}
Lemniscate_constant
Class of mathematical functions
Intuitively, subharmonic functions are related to convex functions of one variable as follows. If the graph of a convex function and a line intersect at
Subharmonic_function
Continuous function that is not absolutely continuous
In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. It is a notorious counterexample in
Cantor_function
Function in analytic number theory
method for efficient evaluation of the eta function. If d k = n ∑ ℓ = 0 k ( n + ℓ − 1 ) ! 4 ℓ ( n − ℓ ) ! ( 2 ℓ ) ! {\displaystyle d_{k}=n\sum _{\ell =0}^{k}{\frac
Dirichlet_eta_function
Theorem in algebraic number theory relating p-adic L-functions and ideal class groups
main conjecture of Iwasawa theory is a deep relationship between p-adic L-functions and ideal class groups of cyclotomic fields, proved by Kenkichi Iwasawa
Main conjecture of Iwasawa theory
Main_conjecture_of_Iwasawa_theory
Complex-valued smooth functions of the upper half plane (harmonic analysis topic)
smooth functions of the upper half plane, which transform in a similar way under the operation of a discrete subgroup Γ {\displaystyle \Gamma } of S L 2 (
Maass_wave_form
the coefficient of the leading term in the Taylor expansion of an Artin L-function associated with a Galois extension K/k of algebraic number fields. The
Stark_conjectures
the 1960s meant that Hasse–Weil L-functions could be regarded as Artin L-functions for the Galois representations on l-adic cohomology groups. Bad reduction
Glossary of arithmetic and diophantine geometry
Glossary_of_arithmetic_and_diophantine_geometry
Warren at MIT. The Patterson function is defined as P ( u , v , w ) = ∑ h , k , ℓ ∈ Z | F h , k , ℓ | 2 e − 2 π i ( h u + k v + ℓ w ) . {\displaystyle P(u
Patterson_function
Hardy–Littlewood maximal function. They play an important role in understanding, for example, the differentiability properties of functions, singular integrals
Maximal_function
Growth curve model
The von Bertalanffy growth function (VBGF), or von Bertalanffy curve, is a type of growth curve for a time series and is named after Ludwig von Bertalanffy
Von_Bertalanffy_function
Dirichlet beta function Dirichlet L-function Hurwitz zeta function Legendre chi function Lerch transcendent Polylogarithm and related functions: Incomplete
List of mathematical functions
List_of_mathematical_functions
German mathematician (born 1958)
Deninger's research focuses on arithmetic geometry, including applications to L-functions. Deninger obtained his doctorate from the University of Cologne in 1982
Christopher_Deninger
Mathematical theorem
theorems of global class field theory. It can be used to prove that Artin L-functions are meromorphic, and also to prove the Chebotarev density theorem. Two
Artin_reciprocity
Function which is integrable on its domain
importance of such functions lies in the fact that their function space is similar to p-integrable function spaces ( L p {\textstyle L^{p}} spaces), but
Locally_integrable_function
Constants of the mathematical zeta function
In mathematics, the Riemann zeta function is a function in complex analysis, which is also important in number theory. It is often denoted ζ ( s ) {\displaystyle
Particular values of the Riemann zeta function
Particular_values_of_the_Riemann_zeta_function
Conjecture in the representation theory of Lie groups
_{n}\boxtimes \mathrm {std} _{n-1})} where L E {\displaystyle L_{E}} is the global L-function obtained as the product of local L-factors given by the local Langlands
Gan–Gross–Prasad_conjecture
Mathematical function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: ψ ( z ) = d d z ln Γ ( z ) = Γ ′ ( z ) Γ ( z )
Digamma_function
Family of solutions to related differential equations
Bessel functions are a class of special functions that commonly appear in problems involving wave motion, heat conduction, and other physical phenomena
Bessel_function
Result of repeatedly applying a mathematical function
function is fed again into the function as input, and this process is repeated. For example, on the image on the right: L = F ( K ) , M = F ∘ F ( K )
Iterated_function
The function l is then an integer-valued function of W; it is a length function of W. It follows immediately from the definition that l(w−1) = l(w) and
Length of a Weyl group element
Length_of_a_Weyl_group_element
Function with a repeating pattern
A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which are used to describe waves
Periodic_function
Seven mathematical problems with a US$1 million prize for each solution
the conjecture is that, if the elliptic curve E has rank r, then the L-function L(E, s) associated with it vanishes to order r at s = 1. The Hodge conjecture
Millennium_Prize_Problems
Objects extending the notion of functions
In mathematics, generalized functions are objects extending the notion of functions on real or complex numbers. There is more than one recognized theory
Generalized_function
Topics referred to by the same term
be assigned In number theory, the value of an L-function In space physics, the value assigned to an L-shell, a particular set of planetary magnetic field
L-value
Formulation of classical mechanics using momenta
mechanics defines the energy function E L ( q , q ˙ , t ) = def ∑ i = 1 n q ˙ i ∂ L ∂ q ˙ i − L . {\displaystyle E_{\mathcal {L}}({\boldsymbol {q}},{\boldsymbol
Hamiltonian_mechanics
Mathematical function characterizing set membership
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all
Indicator_function
Theorem in analytic number theory
class of modified zeta functions and Dirichlet L-functions that possess exactly the same non-trivial zeros as the Riemann zeta function, but whose Euler products
Grosswald–Schnitzer_theorem
Statistical function that defines the quantiles of a probability distribution
the quantile function of a probability distribution is the inverse of its cumulative distribution function. That is, the quantile function of a distribution
Quantile_function
Set of functions between two fixed sets
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is
Function_space
L FUNCTION
L FUNCTION
Male
Swedish
Swedish form of Greek Paulos, PÃ…L means "small."
Boy/Male
Indian, Sanskrit
Miner; L Digger
Male
Hungarian
Hungarian form of Greek Paulos, PÃL means "small."
Male
French
French form of Hebrew Rephael, RAPHAËL means "healed of God" or "whom God has healed."
Boy/Male
Irish
Rooster.
Girl/Female
African, Arabic, Australian, Danish, German, Muslim, Pashtun, Swahili
Pure; L; Holy; Clean; Dean
Male
Scottish
Scottish form of Latin Paulus, PÀL means "small."
Male
Irish
Irish Gaelic form of Greek MichaÄ“l, MÃCHEÃL means "who is like God?"
Male
Norwegian
Norwegian variant form of Scandinavian Njal, NJÃ…L means "champion."
Male
Irish
Irish form of Greek Paulos, PÓL means "small."
Male
Hungarian
Hungarian form of Roman Latin Cornelius, KORNÉL means "of a horn."
Boy/Male
Muslim
Lord of majesty and generosity
Male
Dutch
, God's judge.
Girl/Female
Muslim
Pl of hazz, Fortune, Good l
Male
French
French form of Greek Ioel (Hebrew Yowel), JOËL means "Jehovah is God" or "to whom Jehovah is God."
Girl/Female
Assamese, British, Gujarati, Hindu, Indian, Kannada, Malay, Malayalam, Marathi, Mythological, Oriya, Sindhi, Tamil
Like a Goddess; Daughter of Shukraacharya; L
Male
French
French name derived from Latin natalis dies, NOËL means "day of birth."
Boy/Male
Indian
Lord of majesty and generosity
Girl/Female
Indian
Pl of hazz, Fortune, Good l
Male
French
Masculine form of French Gaëlle, GAËL means "holy and generous."
L FUNCTION
L FUNCTION
Boy/Male
Hindu, Indian
Powerful; Delightful
Female
English
French form of German Adala, ADELLE means "noble."
Girl/Female
Norse
Heroic.
Girl/Female
Hindu
She is pleasing like the Moon
Boy/Male
British, English
From the Old Ford
Boy/Male
Muslim
Inhabitant
Boy/Male
Indian
Pureness, Pure, Precious
Girl/Female
Assamese, Gujarati, Hindu, Indian, Kannada, Marathi, Sindhi, Tamil, Telugu
Honest
Boy/Male
Tamil
Line
Male
Finnish
Finnish name AIMO means "generous amount."Â
L FUNCTION
L FUNCTION
L FUNCTION
L FUNCTION
L FUNCTION
n.
A weed of the genus Lamium (L. amplexicaule) with deeply crenate leaves.
a.
Relating to Casserio (L. Gasserius), the discover of the Gasserian ganglion.
n.
An imperfect enunciation of the letter r, in which it sounds like l.
n.
A large stork of the genus Leptoptilos (formerly Ciconia), esp. the African species (L. crumenifer), which furnishes plumes worn as ornaments. The Asiatic species (L. dubius, or L. argala) is the adjutant. See Adjutant.
n.
The name of the Greek letter /, /, corresponding with the English letter L, l.
n.
A short right-angled pipe fitting, used in connecting two pipes at right angles.
v. t.
To betray; to show. [L.]
n.
Any small leguminous plant of the genus Lathyrus, especially L. Nissolia.
n.
An extension at right angles to the length of a main building, giving to the ground plan a form resembling the letter L; sometimes less properly applied to a narrower, or lower, extension in the direction of the length of the main building; a wing.
n.
See L.
L. catechunenus, Gr.
One who is receiving rudimentary instruction in the doctrines of Christianity; a neophyte; in the primitive church, one officially recognized as a Christian, and admitted to instruction preliminary to admission to full membership in the church.
n.
A symbol representing fifty units, as 50, or l.