Here are two images to complement joro's image.

Zeroes of $ \zeta(s)^2 - \zeta(1-s)^2 $:

Zeroes of $\left ( 2^s \pi^{s−1} \sin \left ( \frac{\pi s}{2} \right ) \Gamma (1-s) \right ) ^2 - 1$:

In particular it seems to be the case that all zeroes inside the critical strip are on the critical line, and that all other zeroes don't stray too far from the origin, as hinted at by GH's answer.