# Real values of $\frac{\zeta(2 s)}{\zeta(s)}$

If $$\frac{\zeta(2 s)}{\zeta(s)}$$ is a real number, then must $$s$$ be real ?

Of course not. For example, let $$2s$$ be any non-real zero of the zeta function where $$s$$ is not a zero.
Through such a point, there will be a curve along which $$\zeta(2z)/\zeta(z)$$ is real.
EDIT: Here is a plot of the curve where $$\zeta(2s)/\zeta(s)$$ is real in one region of the plane. A dot indicates the one non-real point in this region where $$\zeta(2s) = 0$$.
• What about it? Everywhere on the curve except at $s$ where $\zeta(2s)=0$, the ratio is real and nonzero. – Robert Israel May 1 at 14:53