L-function of a cusp form ($L(f,s)={\sum }_{n\ge 1}{a}_n{n}^{-s}$)

For $f={\sum }_{n\ge 1}{a}_n{q}^n$, define $L(f,s)={\sum }_{n\ge 1}{a}_n{n}^{-s}$ 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.