I am curious about the definition of the $\zeta(s)$. Of course the $p$-series definition does not converge for $Re(s)<1$, so analytic continuation is required. Previously I have read that the functional equation is applicable to the whole complex plane except $s \neq 1$. Naively however, it does not seem to work on the critical strip. E.g., $\zeta(1/3) = ... \times \zeta(2/3)$, but $\zeta(2/3)$ would invoke $\zeta(1/3)$ and we have an issue. Similarly, $\zeta(3) = ... \times \zeta(-2)$, which would mean that $\zeta(2n+1) = 0, n \in \mathbb{N}$, but that is certainly not true.
Hence, I wanted to know whether it was appropriate to define it as,
$ \zeta(s) = \begin{cases} \sum_{n=1}^{\infty} \frac{1}{n^s} & Re(s) > 1 \\ \frac{1}{1-2^{1-s}} \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^s} & 0 < Re(s) < 1 \\ 2^s \pi^{s-1} \sin(\frac{\pi s}{2}) \Gamma(1-s) \zeta(1-s) & Re(s)<0 \end{cases} $