I need to plot the following functional with accuracy:
$$ I(x,s) =\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy,s) − F(x −\mathrm iy,s)}{\mathrm e^{2πy}-1}, $$ Where $ F(z,s) = \dfrac{1}{z^s\Gamma(\sin^2[π\Gamma(z)/(2z)])} $.
And let us restrict $s\in[0,1]$
What is the most efficient way of computing this integral?
Nature of functional as $x\rightarrow\infty$ from the computation ?
I computed some relatively small values using Mathematica which suggests the function is oscillatory with damping. But I need big values greater than x=100 and at s=1