For $\Re(s)>1$, the Riemann zeta function $\zeta(s)$ is defined by
$$\zeta(s) = \sum_{n=1}^\infty n^{-s},$$
or equivalently by the Euler product
$$\zeta(s) = \prod_{p} (1-p^{-s})^{-1}, $$
where $p$ runs over the entire set of primes. My question is, what is the Taylor series of $\log \zeta(s)$ ?