Laurent series expansions of $L$-functions

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.