Consider the following sum :

$$S(p,s)=\sum_{n=1}^\infty\frac{\sin^2(\frac{p\Gamma(n)}{n})}{n^s}$$

Here , $p$ is a variable w.r.t which we are going to analyse the sum

$s\in(0,1]$

I tried to use Abel - Plana summation formula(ABSF) for partial sum :

we can get some crude estimates on integral :

$$I_1(q)= \int_1^q\frac{\sin^2(\frac{\Gamma(t)}{t})}{t^s}dt $$

As $q\rightarrow \infty $

( Integrand oscillates very wildly in right half plane )

But it seems second integral in ABSF is impregnable .

Question : Can we get divergence criteria for the sum w.r.t parameters $p$ and $s$?

Note : Second integral in ABSF:

$$I_2(x)=\int_0^\infty \frac{F(x + iy,s) − F(x − iy,s)}{e^{2πy}-1}dy$$

Where , $F(z)=\frac{\sin^2(\frac{\Gamma(z)}{z})}{z^s} $

I tried to get estimate on this as $x\rightarrow \infty $ but in vain .