Editing L-function (section)

{{Short description|Meromorphic function on the complex plane}}
{{Expand German|L-Funktion|date=March 2024}}
{{DISPLAYTITLE:''L''-function}}
[[File:Riemann-Zeta-Func.png|right|thumb|300px|The [[Riemann zeta function]] can be thought of as the archetype for all ''L''-functions.<ref>{{cite web |first=Jörn |last=Steuding |title=An Introduction to the Theory of ''L''-functions |work=Preprint |date=June 2005 |url=https://www.scribd.com/document/230217684/An-Introduction-to-the-Theory-of-L-Functions }}</ref>]]

In mathematics, an '''''L''-function''' is a [[meromorphic]] [[Function (mathematics)|function]] on the [[complex plane]], associated to one out of several categories of [[mathematical object]]s. An '''''L''-series''' is a [[Dirichlet series]], usually [[convergence (mathematics)|convergent]] on a [[half-plane]], that may give rise to an ''L''-function via [[analytic continuation]]. The [[Riemann zeta function]] is an example of an ''L''-function, and some important conjectures involving ''L''-functions are the [[Riemann hypothesis]] and its [[Generalized Riemann hypothesis|generalizations]].

The theory of ''L''-functions has become a very substantial, and still largely [[conjectural]], part of contemporary [[analytic number theory]]. In it, broad generalisations of the Riemann zeta function and the [[Dirichlet L-function|''L''-series]] for a [[Dirichlet character]] are constructed, and their general properties, in most cases still out of reach of proof, are set out in a systematic way.  Because of the [[Euler product formula]] there is a deep connection between ''L''-functions and the theory of [[prime number]]s.

The mathematical field that studies ''L''-functions is sometimes called '''analytic theory of ''L''-functions'''.