It is easy to show that the for the series representing the Riemann zeta function for $\Re(s)>1$

$$\zeta(s)=\sum\frac{1}{n^s}$$

the symmetry $\zeta(\overline{s})=\overline{\zeta(s)}$ holds.

**Question: Does this symmetry hold for any analytic continuation of this series? If so, why?**

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For example, by doing the laborious algebra, we can show the symmetry also holds for the following continuation valid for $\Re(s)>0$

$$\zeta(s)=\frac{1}{1-2^{1-s}}\sum\frac{(-1)^{n+1}}{n^{s}}$$

I would like to state that the symmetry holds by referring to a general principle of complex functions - which I don't know.