Laurent series expansions of $L$-functions

TL;DR

This paper revisits Laurent series expansions of key L-functions at s=1 and introduces a new Laurent series expansion for L-functions related to cusp forms over the full modular group.

Contribution

It provides a comprehensive analysis of Laurent series expansions at s=1 for classical L-functions and introduces a novel expansion for cusp form L-functions.

Findings

Revisited Laurent expansions for zeta and Dirichlet L-functions at s=1

Derived a new Laurent series expansion for cusp form L-functions

Enhanced understanding of the local behavior of L-functions near s=1

Abstract

One of the main objectives of the current paper is to revisit the well known Laurent series expansions of the Riemann zeta function $\zeta(s)$, Hurwitz zeta function $\zeta(s,a)$ and Dirichlet $L$-function $L(s,\chi)$ at $s=1$. Moreover, we also present a new Laurent series expansion of $L$-functions associated to cusp forms over the full modular group.