Zeta function Re(s)=1

Question

Here I am considering the original zeta function (not the extended one)$$\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s}$$ When Re(s)>1, the Zeta function converges, and if Re(s)<1 it diverges.

Here is my question. What happens when Re(s)=1?

I know it diverges when s=1, but does it converge otherwise?