$J(z)= \frac{1}{\pi} \int _0 ^\infty \frac{z}{\eta^2+z^2} \ln (\frac{1}{1-e^{-2\pi \eta}}) d \eta$ is analytic in Re $(z)>0$

Question

In the Complex Analysis book of Ahlfors, the Stirling formula is written as follows:

Stirling's Formula. $\Gamma(z)= \sqrt {2 \pi} z^{z-1/2} e^{-z}e^{J(z)}$ (Re$ z>0$, where $J(z)= \frac{1}{\pi} \int _0 ^\infty \frac{z}{\eta^2+z^2} \ln (\frac{1}{1-e^{-2\pi \eta}}) d \eta$.

My question is: Is $J(z)$ analytic in Re$z>0$? I know that $J(z) \to 0$ as $z \to \infty$ (in Re$z>0$), but I'm not sure that $J$ is analytic in the right half plane.

Can we show that $J$ is analytic, by for example Morera's theorem or something eles?