I have been reading about the RH recently and I understood most of it until now. However, the biggest problem I'm having is to know what are the forms of the Riemann zeta function for the 3 main regions in the complex plane, $\Re(s) <-1$, $0 \le \Re(s) < 1$, and for $\Re(s)> 1$. Also, I have seen that zeta can be defined as the following integral.
$$ \frac{1}{\Gamma(s)}\int_0^\infty \frac{x^{s-1}}{e^x-1}\, \mathrm{d}x,$$
Is the zeta function defined on the entire complex plane, except $1$? And about the other ones? Also, are there other integrals for zeta, some whose limits of integration are different than zero and infinity?
From the picture in the youtube video by numberphile starting at 11:44
$$\zeta(s)=\sum\limits_{n=1}^{\infty}\frac{1}{n^s}=\prod\limits_{p \text{ prime}}\frac{p^s}{p^s-1}, \;\;\;\;\; \Re(s)>1$$
$$\zeta(s)=(1-2^{1-s})\sum\limits_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^s}, \;\;\;\;\; 0<\Re(s)<1$$
$$\zeta(s)=\left((2^s\pi^{s-1}\sin\left(\frac{\pi s}{2}\right)\Gamma(1-s)\right)\zeta(1-s),\;\;\; \Re(s)<0$$
There is this integral relation in the OEIS: $$\left(1-\frac{1}{2^{s-1}}\right) \zeta (s) \Gamma (s+1)=\int_0^{\infty } \frac{1}{e^{x^{1/s}}+1} \, dx, \;\;\;\; \Re(s)>0$$