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

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