= L-function of a cusp form
{c}
{title2=$L(f,s)=\sum_{n\geq1}a_nn^{-s}$}
= L-functions of cusp forms
{c}
{synonym}
For $f=\sum_{n\geq1}a_nq^n$, define $L(f,s)=\sum_{n\geq1}a_nn^{-s}$ in a right half-plane. Its completion $(2\pi)^{-s}\Gamma(s)L(f,s)$ is the <Mellin transform> of $f(iy)$. The modular inversion and exponential cusp decay continue this completion to an entire function and give its reflection $s\mapsto k-s$. A normalized <Hecke eigenform> also has an <Euler product of a Hecke eigenform>.
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