I have a Fourier transform to compute, or at least to estimate, and I do not have any idea of how to deal with it. Let $f = \frac{\Gamma'}{\Gamma}$ be the diagmma function, and $g(x) = \exp(-2f(\frac{1}{4}+i\frac{x}{2}) -2 f(\frac{1}{4}-i\frac{x}{2}))$. I want to compute its Fourier transform $$\hat{g}(y) = \int_{\mathbb{R}} g(x) \exp(2i\pi xy) dx$$
I am a bit stuck with that, I don't see how to use change of variables or contour shifting to get something more handable. I will be glad to explore any clue or idea!