For fixed $h>0$, consider the function
$$F(x) = e^{-x\pi/2} \left| \Gamma\left(h-i\frac x2\right)\right|^2.$$
What is its asymptotic expansion as $x\to 0$ or $x\to \pm\infty$?

For $x\to \pm\infty$, I only need the leading behavior, while for $x\to 0$, I need the first subleading term that depends on $x$.

I tried Mathematica, but it does not provide a useful answer.