Plot of a function defined by an integral

Question

(First of all, this is my first Mathematica question. I'm not used to Mathematica that much. So, apologies in advance.)
I need to plot the following functional with accuracy:
$$
I(x,s) =int_0^inftymathrm dy frac{F(x + mathrm iy, s) − F(x −mathrm iy, s)}{mathrm e^{2πy}-1},
$$
Where $ F(z, s) = dfrac{sin^2[πGamma(z)/(2z)]}{z^s} $.
And let us restrict $sin[0,1]$
Also, can we get quantitative upper and lower bound estimations on the functional using Mathematica?
The reason for the question is that the functional gives very massive values ( upto 10^100) after the value x=6 which I think are not correct. I don't know how to resolve this issue.So I'm posting this question for bigger accurate values

chris · Accepted Answer

You could do it numerically?

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

Then

ParallelTable[{x, II[x, s] // Im}, {s, 1, 2, 1/2}, {x, 1, 5, 0.05}] //
  ListLinePlot[#, PlotRange -> All] &

Plot of a function defined by an integral

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