In layman terms, What does the functional equation $\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{\pi s}{2}\right)\Gamma(1-s)\zeta(1-s)$ states in relation to the analytical continuation of the Euler-Riemann zeta function $\zeta(s)=\sum\limits_{n=1}^\infty\dfrac{1}{n^s}$, $\Re s>1$?
What is the explanation?
Thanks in advance.
The functional equation relates $\zeta(s)$ and $\zeta(1-s)$. Euler's definition of $\zeta(s)$ converges for $\Re s>1$ only, i.e. $s=1+a+it$ where $a>0$. So the functional equation relates $\zeta(1+a+it)$ and $\zeta(1-(1+a+it))=\zeta(-a-it)$. Since $a>0$ then $-a<0$. So using Euler's definition of $\zeta(s)$, the functional equation evaluates the $\zeta$ function whose argument has negative real part. You end up with analytic continuation of $\zeta(s)$ to the entire complex plane except for the strip $0\leq \Re s\leq 1$.
Going beyond, it's possible to manipulate Euler's definition into an infinite sum or an integral which is valid inside the critical strip.