Mellin continuation of a noncuspidal modular form (${\Lambda }_f(s)=I_f(s)+i^k{I}_f(k-s)+a_0\left(\frac{i^k}{s-k}-\frac{1}{s}\right)$)

For a level-one modular form of positive even weight $k$ and constant coefficient $a_0$, put $I_f(s)={\int }_1^{\infty }(f(iy)-a_0)y^{s-1}dy$. This is an entire function. Splitting the Mellin transform at one and using $f(iy)=i^k{y}^{-k}f(i/y)$ proves the displayed continuation of $(2\pi {)}^{-s}\Gamma (s)L(f,s)$. It obeys ${\Lambda }_f(s)=i^k{\Lambda }_f(k-s)$ and has possible simple poles only at zero and $k$, with residues $-a_0,i^k{a}_0$. For a cusp form both disappear.