A particular case of Parseval's theorem for Fourier transforms says that if $f$ is square integrable on $\mathbb{R}$, then
$$ \int_{-\infty}^{\infty} |f(t)|^{2} \ dt = \int_{-\infty}^{\infty} |\hat{f} (\omega)|^{2} d \ \omega .$$
I recall coming across a similar theorem for Mellin transforms that states under certain conditions,
$$ \int_{0}^{\infty} \frac{|f(x)|^{2}}{x} \ dx = \frac{1}{2 \pi}\int_{-\infty}^{\infty} |F(it)|^{2} \ d t$$
where $F(s)$ is the Mellin transform of $f(t)$.
Using this theorem we can evaluate an integral like $ \displaystyle \int_{-\infty}^{\infty} \Gamma(a+it) \Gamma(a-it) \ dt$ fairly easily.
But I can't find much information about this theorem on the internet.
Is this somehow just a corollary of the other theorem?