Just a remark, this is true for any two element sequence (no, it's not infinite). 

For example

$$1 + \frac{1}{2^s}.$$ All zeros have real part zero.

A less obvious experimental fact is that the same (the zeros lie on a vertical line, not zero) is true for

$$1 + \frac{1}{2^s} + \frac{1}{3^s}.$$

Here is the picture from mathematica (contour lines of $|f| = 0.2,$ in case you are wondering).

[![enter image description here][1]][1]

The next picture is what happens when you have the $\sum_{k=1}^{10} k^{-s}.$
[![enter image description here][2]][2]


  [1]: https://i.sstatic.net/13XCJ.png
  [2]: https://i.sstatic.net/URGr0.png