TITLE: How to prove $\zeta'(0)/\zeta(0)=\log(2\pi)$?
QUESTION [9 upvotes]: How do I prove that $\zeta'(0)/\zeta(0)=\log(2\pi)$ ?
I can get $\zeta(0)=-\frac{1}{2}$, but I don't know how to calculate $\zeta'(0)=-\frac{1}{2}\log(2\pi)$ ? Can you help me ?
Here $\zeta(s)$ is Riemann zeta function:
$$\zeta(s):=\sum_{n=1}^{\infty}\frac{1}{n^s}. $$

REPLY [2 votes]: Maybe you're interested to check $(38)$ here. 
The Wallis formula may also be written as 
$$\left(4^{\zeta{(0)}} \cdot e^{-\zeta'{(0)}}\right)^2=\frac{\pi}{2}$$
Chris.