The Riemann Zeta Function can be analytically continued to the entire complex plane into the functional equation:

(1) \zeta(s)=2^s\pi^{s-1}\sin{\left(\frac{\pi{s}}{2}\right)}\Gamma(1-s)\zeta(1-s)

I've searched the net for many hours trying to see a demonstration of how this
continuation occurs, but alas I have found none. If someone could show this or
point me in the direction of somewhere that can, I'd be most gracious.

Also, I would love it if someone who is knowledgable on the Riemann Zeta Function and the Riemann Hypothesis to give a short lecture on it perhaps? CaptainBlack maybe?

Thanks guys.