Ah, so it's only applicable in this special situation. I thought there was some bigger machinery running in the background such as a UCT with local coefficients. Thank you for the quick response!

Sorry for excavate this thread, but could you elaborate the step where you use UCT? I couldn't find any evidence that UCT still holds in the case of local coefficients. The literature is unfortunately very sparse on this.

I wonder what goes wrong in the non-smooth case?

The very last comment did answer my first question partially. Do you have a source on that?

Yeah sure. Giving it a second thought, there should be no problem. Usually, quantfiers are unbounded anyway. But the second question keeps still there and is worth thinking about, I guess.

The final result actually fits. Apparently it's a misprint.

Found the mistake. The wanted residues are twice the residues of the function above. It was totally my mistake, didn't see it. The comments can be deleted. Thanks for your detailed help.

@HenriCohen: How did you get your result for $s=\frac{3}{2}$? The integral doesn't seem so trivial for me as I thought first. Maybe there are some factors missing? By the way, I missed a factor of 2 above. But that doesn't solve the problem.