I am trying to compute the double integral for fixed $ m,z>0$ :

` Integrate[(Gamma[y/2] Sqrt[Gamma[3 - y]/Gamma[y]])/ Gamma[(3 - y)/2] z^(3 - y) (Exp[-m x] - 1) x^(y - 3)/x, {x, 0, \[Infinity]},{y, 3/2 - I \[Infinity], 3/2 + I \[Infinity]}] `

The integral over $ x$ can be done analytically, and the result depends on the product $ mz$ , so there is effectively just one parameter. The $ x$ integral needs to be split into two regions I suspect, and in one region the contour of the $ y$ integral would need to be deformed so that it remains convergent. Since the $ y$ integral involves complicated branch cuts, I wanted to be able to do it numerically for a range of $ mz$ , to get a least a few digits of precision. I am having difficulty getting stable results numerically tho.