The Riemann zeta function for complex conjugates

Question

If $\zeta(x)=a+ib$ and $\zeta(y)=a-ib$, is there a single equation that relates $\zeta(x)$ to $\zeta(y)$?

Answer 1

Indeed, it is true that any meromorphic function $f(z)$ that is real on the real axis satisfies $f(\bar z) = \overline{f(z)}$. This fact is a combination of the Schwarz reflection principle and the uniqueness of analytic continuations.

(Even more pedantically: all that is required is that $f(z)$ is real on a subset of the real axis with a limit point inside the domain of analyticity. I mention this because the simplest way to see that $\zeta(s)$ is real on the real axis is via the Dirichlet series $\zeta(s) = \sum_{n\in\Bbb N} n^{-s}$, which is valid on the interval $(1,\infty)$.)