Consider the complex gamma function, denoted by $\Gamma(\sigma+it)$.

Now, let's fix $\sigma$ and let t vary. Then consider the following expression:

$$|\Gamma(\sigma+it)|^2$$

For any choice of $\sigma$ such that $\Gamma(\sigma)$ isn't a pole, this will appear to be (almost) a two-sided probability density function, save for that it isn't normalized. It decays around as quickly as $e^{-|t|}$, somewhat resembles the function $e^{-\sqrt{1+t^2}}$, and appears related to the hyperbolic distribution.

Is there a precise closed-form expression for this function in terms of elementary or Louvillian functions?

Is there a name for this probability distribution (assuming it's normalized)?

This question was originally asked on MSE here. It got some attention but no answers; someone commented I should repost here.