How to prove $\zeta'(0)/\zeta(0)=\log(2\pi)$?

Question

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: $$\zeta(s):=\sum_{n=1}^{\infty}\frac{1}{n^s}. $$

Answer 1

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.

Answer 2

Begin with $$ \zeta(1-z) = 2 (2\pi)^{-z}\cos\frac{\pi z}{2} \Gamma(z)\;\zeta(z) $$ then take logarithmic derivative. Can you finish?