We have the integral :

$$\int_{0}^{\infty}\log\left(1+\frac{s^{2}}{4\pi^{2}} \log^{2}(1+ix)\right ) e^{-2\pi nx}dx$$

Where s is a complex parameter, and n is a positive integer. The integral converges by virtue of the exponential factor. I tried to deform the path of integration such that we avoid the branch cut(s) of the logarithm. But here is where i got stuck, the internal complex log makes it confusing to do so !

A different version of this problem was posted here with no answers.