Is Riemann Zeta Function symmetrical about the real axis?

Question

From wikipedia, http://en.wikipedia.org/wiki/Riemann_zeta_function

"Furthermore, the fact that $\zeta(s) = \zeta(s^*)^*$ for all complex s ≠ 1 ($s^*$ indicating complex conjugation) implies that the zeros of the Riemann zeta function are symmetric about the real axis."

I know the zeros are symmetrical. But what about the other values of $\zeta(s)$? My main aim is to find out:

Is $\zeta(s)$ symmetrical about the real axis for all $\Re(s) > 1$ ?

Answer 1

Let $f$ be a holomorphic function with $f(\overline{z}) = f(z)$, then $f$ is necessarily a real constant function, so the most you can ask for is $\zeta( \overline{z}) = \overline{\zeta(z)}$, since the Riemann zeta function is not a constant function.