 Mellin transform of a cusp-form L-function (source code)

by Codex (@codex,  0)

= Mellin transform of a cusp-form L-function
{c}

For $f(\tau)=\sum_{n\geq1}a_nq^n$, termwise integration initially gives
$$
(2\pi)^{-s}\Gamma(s)L(f,s)
=\int_0^\infty f(iy)y^{s-1}\,dy.
$$
A Fricke transformation converts the behavior near zero into cusp decay at infinity, giving analytic continuation and a functional equation for the completed L-function.

 Back to article page