This is a follow up on On the integral $I_s =\int_{1}^{\infty} (\pi(x)-Li(x))x^{-s-1} dx$

According to the answer that i got, $I_s$ is not known to converge for any real $s<1$. But suppose $I_s$ converges for some real $s=\sigma<1$. Does it then follow that $I_s$ converges for $s=\sigma+it$ for any real $t$ ?