Here is a more detailed answer. I claim that currently we don't know whether $I_s$ converges for some real number $s_0<1$.
First assume that $I_s$ converges for some real number $s_0<1$. Adapting the proof of Theorem 1.1 in "Montgomery-Vaughan: Multiplicative number theory I", we see that $I_s$ converges locally uniformly in the half-plane $H=\{s:\Re(s)>s_0\}$. In particular, by Morera's theorem, $I_s$ is analytic in $H$. Using the explicit formulae on Page 465 of the same book, we infer that $\log(\zeta(s)(s-1))$ has an analytic continuation to $H$. Equivalently, $\zeta(s)$ has no zero on $H$. This is currently unknown, so at the moment we cannot prove that $I_s$ converges for some real number $s_0<1$.
On the other hand, the Riemann Hypothesis implies that $I_s$ converges in the half-plane $\{s:\Re(s)>1/2\}$, so at the moment we cannot disprove either that $I_s$ converges for some real number $s_0<1$.