sequences and series – Method to evaluate an infinite sum of ratio of Gamma functions (how does Mathematica do it?)

This question arose from Amdeberhan’s question, the evaluation of a double integral, which can be reduced to the evaluation of this series:
$$sum _{n=0}^{infty } frac{Gamma left(n+frac{1}{2}right)^2 Gamma left(n+frac{s}{2}right)}{Gamma (n+1)^2 Gamma (n+s)}=frac{pi ^2 2^{1-s} Gamma left(frac{s}{2}right)}{left(Gamma left(frac{3}{4}right) Gamma left(frac{s}{2} +frac{1}{4}right)right)^2},;;{rm Re},s>0.$$
The evaluation of the sum is Mathematica output. Can someone enlighten me as to how this calculation proceeds?

I went so far as to pay for Wolfram Alpha Pro, hoping that it would disclose the steps, but to no avail. What is even more frustrating is that for $s=1$ the right-hand-side is the square of a complete elliptic integral, which is also recognized immediately by Mathematica and was the original question in the cited post, so far without a conclusive answer.