In all proofs of the Weil-Guinand explicit formula, there's this step (this is from [Paul Garrett's notes][1]):

[![enter image description here][2]][2]

Now consider this: 

(1) $\frac{\zeta^\prime(s)}{\zeta(s)}$ has poles at $s=1$ and $s=\frac{1}{2} + i \gamma$, so if we move the contour from $\Re(s) = 1+\epsilon$, to $\Re(s) = \frac{1}{2}$, we'll pick up all the residues.

(2) Yet if we write out the Dirichlet series of $\frac{\zeta^\prime(s)}{\zeta(s)}$, then $p^{-ms}$ has no pole between $\Re(s) = 1+\epsilon$ and $\Re(s) = \frac{1}{2}$, so the proof claim we can move the contour to $\Re(s) = \frac{1}{2}$ without picking up any residues. Is this really justified?

So the question is: does moving the contour involve any residues? (1) says yes. (2) says no. This is strange. 

  [1]: http://www.math.umn.edu/~garrett/m/mfms/notes_2015-16/04_Guinand_explicit_fml.pdf
  [2]: https://i.sstatic.net/LqBon.png