L-function

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

In it, broad generalisations of the Riemann zeta function and the 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.

The general constructions start with an L-series, defined first as a Dirichlet series, and then by an expansion as an Euler product indexed by prime numbers.

Then one asks whether the function so defined can be analytically continued to the rest of the complex plane (perhaps with some poles).

In the classical cases, already, one knows that useful information is contained in the values and behaviour of the L-function at points where the series representation does not converge.

One of the influential examples, both for the history of the more general L-functions and as a still-open research problem, is the conjecture developed by Bryan Birch and Peter Swinnerton-Dyer in the early part of the 1960s.