My question is how to evaluate Riemann's functional equation for values $\Re(s)<0$?

I know I must use the functional equation which is defined as $\zeta (s)=2^s\pi {}^{s-1}\sin \left(\frac{\pi s}{2}\right)\Gamma (1-s)\zeta (1-s)$.

Say I wanted to find $\zeta (-3+2i)$. I know how to calculate $2^s\pi {}^{s-1}\sin \left(\frac{\pi s}{2}\right)\Gamma (1-s)$ at $-3+2i$, more specifically my question is about the $\zeta (1-s)$ term. Would I use the sum formula of the zeta function since $1-(-3+2i)=4-2i$ and thus $\Re(s)>1$?

Thanks for any help received.

  • 1
    $\begingroup$ Yes, indeed, you can use $\zeta(s)=\sum_n n^ {-s}$ for all $s\in \mathbb{C}$ with $Re(s)>1$. $\endgroup$ – Dietrich Burde Mar 9 '18 at 8:36

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