# Asymptotic of a functional as $x\rightarrow \infty$

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 $$F(z, s) = \dfrac{\sinh(\sin^2[π\Gamma(z)/(2z)])}{z^s}$$.

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

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 ?

Not trivial : see:

https://math.stackexchange.com/q/3570663/702232

Construction of a certain weight for a functional to satisfy given condition: