Numerical estimates on asymptotic of given functional as $x\rightarrow \infty$

Question

Consider the following functional :

$$ I(x,s) =\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy, s) − F(x −\mathrm iy, s)}{\mathrm e^{2πy}-1}, $$ where $ F(z, s) = \dfrac{\sin^2[π\Gamma(z)/(2z)]}{z^s} $.

Let us restrict $s\in[0,1]$

In Mathematica

f[z_, s_] = Sin[π Gamma[z]/(2z)]^2/z^s; 
II[x_, s_] := Integrate[(f[x + I y, s] - f[x - I y, s])/(Exp[2 Pi y] - 1), 
             {y, 0,Infinity}]

Can we get sharp numerical asymptotic of $I(x)$ as as $x\rightarrow \infty$?

Also, can we get quantitative upper and lower bound estimations on the functional ?

Update : Also try for $$F(z)= \log(z){\sin^2[π\Gamma(z)/(2z)]}$$