Say I have a good estimate for the $L^2$ mean of the Riemann zeta function $\zeta(s)$ for $\Re s = 1/2$, $|t|\leq T$:
$$\int_0^T |\zeta(1/2+i t)|^2 = T \log T - T (1 + \log 2 \pi - 2\gamma) + O(T^\alpha).$$
for some $\alpha\in (0,1/2]$. Does a good estimate on $$\int_0^T |\zeta(\sigma+it)|^2$$ follow, for $\sigma<1$, $\sigma\ne 1/2$? What would its error term be?