Generalized L-functions for meromorphic modular forms and their relation to the Riemann zeta function

Abstract

In this paper, we construct a family of generalized $L$-functions, one for each point $z$ in the upper half-plane. We prove that as $z$ approaches $i\infty$, these generalized $L$-functions converge to an $L$-function which can be written in terms of the Riemann zeta function.