# generalizing L-functions and modular forms

A Dirichlet L-function looks like this:

$L(\chi,s)=\sum_{n=1}^{\infty} \frac{\chi(n)}{n^{s}}$

http://en.wikipedia.org/wiki/Dirichlet_L-function

And the related modular form looks something like this:

$f(\tau)=c_{0}+\sum_{n=1}^{\infty} \chi(n)e^{2\pi i \tau}$

What about the following:

$G(\chi,\tau,s) = c_{0}+\sum_{n=1}^{\infty} \frac{\chi(n)e^{2\pi i \tau n}}{n^{s}}$

Mathworld, in the above page, mentions that: “Dirichlet L-series can be written as sums of Lerch transcendents with z a power of e^(2pii/k).”

But an explicit google search for the following does not appear to result in much useful things:
“lerch transcendent” “modular forms” Hecke. I may generate a video showing the interpolation of $G(\chi,\tau,s)$ for a given Dirichlet character from L-function to modular form shortly.

But, lo, I spoke too soon: THE LERCH ZETA FUNCTION I. ZETA INTEGRALS