What is the value of the series?:
$$\sum_{k=1}^\infty\frac {\zeta(-k)} k$$
Where $\zeta(z)$ is the Riemann Zeta function and for every negative integer $n$ we have $\zeta(n)=-\frac {B_{n+1}} {n+1}$.
I know the identity:
$$\sum_{k=2}^\infty(-1)^k\frac {\zeta(k)} k=\gamma$$
(where $\gamma$ is the Euler-Mascheroni constant) which is pretty similar so I was thinking if the series I wrote converges too ? Is there any closed form for the result ?