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).
The next picture is what happens when you have the $\sum_{k=1}^{10} k^{-s}.$
