Is it possible to compute the inverse transform of
$$ \frac{1}{a^{-s}\cos( \frac{\pi s}{2})\Gamma (s)} $$
or similarly is it possible to compute the Inverse Mellin transform ??
$$ \frac{ \zeta (1-s)}{\zeta (s)} $$
$$ \frac{ \zeta (s)}{\zeta (1-s)} $$
The Mellin inverse is given by
$$ \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}dsF(s)x^{-s} $$