L-function

An L-function is a meromorphicfunction on the complex plane, and one out of several categories of mathematical objects studied in analytic number theory and related fields. L-functions share fundamental properties and characteristics with the Riemann zeta function, which serves as the prototypical example of an L-function; therefore, L-functions are generalisations of the Riemann zeta function. Some important conjectures involving L-functions are, consequently, the Riemann hypothesis and its generalisations.

A Dirichlet series, usually convergent on a half-plane, that may give rise to an L-function via analytic continuation, is called an L-series.

Fundamental subclasses of L-functions were built on the work of Leonhard Euler (which is now known as the Riemann zeta function). Most notably, the mathematicians Bernhard Riemann (1826–1866), Richard Dedekind (1831–1916), Erich Hecke (1887–1947) and Emil Artin (1898–1962) investigated the subclasses of L-functions, discovering eponymous L-functions each.

The terms "L-function" and "zeta-function" are often used synonymously due to the fundamentally similar and derivative nature of the work, however, not all zeta-functions are L-functions. Most notably, the Prime zeta function is not an L-function, since they cannot be analytically extended to the entire complex plane.