= Mellin continuation of a noncuspidal modular form
{c}
{title2=$\Lambda_f(s)=I_f(s)+i^kI_f(k-s)+a_0\left(\dfrac{i^k}{s-k}-\dfrac1s\right)$}
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 $-a_0,i^ka_0$. For a <cusp form> both disappear.
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