Let
$$F_n(z)=\dfrac\pi4\,(z^2-1)\,z^{\large-\frac{n+4}2}\cot{\pi\dfrac{(z-1)^2}{4z}}$$
I'm looking for a closed form of
$$1+\dfrac1{2\pi i}\oint\limits_C F_n(z)\text dz$$
$$C =\left\{\dfrac{52}{16-y^2}+iy,\;y\in(-4,4)\right\}$$
To be honest I've no idea on how to proceed further. I'm not even sure if there even exists a nice closed form. However I got a hint from someone who said that for $n \geqslant 3$, you can relate this to the Riemann zeta function, in short the closed form contains Riemann zeta function.
Your insight would be very helpful, Thanks in advance.