Oddball Zeta functional equation: uh?

[benorin]

I found this functional equation for the Riemann zeta function in Table of Higher Functions, 6th ed. by Jahnke, Emde, & Losch on pg. 40:

$$z(z+1)\frac{\zeta (z+2)\zeta (1-z)}{\zeta (z)\zeta (-1-z)}=-4\pi ^2$$​

any suggestions as to how one might consider such an equation, much less derive it? There is, for example, an analytic continuation of the zeta function to all points in the complex plane except z=1, see this thread for the particular continuation of which I speak.

 

[shmoe]

It's just the product of one of the more usual functional equations for different values of s to cancel out the gamma factors.

$$\chi(1-s)=\frac{\zeta(1-s)}{\zeta(s)}$$

where

$$\chi(s)=\pi^{s-1/2}\frac{\Gamma(\frac{1-s}{2})}{\Gamma(\frac{s}{2})}$$

so

$$\frac{\zeta(z+2)\zeta(1-z)}{\zeta(-1-z)\zeta(z)}=\chi(z+2)\chi(1-z)$$

Where we've used the usual functional equation once with s=-1-z and once with s=z. The rest is straightforward with the usual functional equation of Gamma.

 

[benorin]

Got it! Thanks shmoe.