Denote by $\zeta$ the Riemann zeta function, and let $\sigma, T \in \mathbb{R}^+$ where $\sigma \geq 1/2$. I'm going through the proof of Theorem 13.8 of Titchmarsh's The theory of the Riemann zeta function. At some point in the proof, it is claimed without proof (presumably because its a trivial result) that the residue of the integral
$$\int_{\Re(s)=5} \frac{\log \zeta(s + iT)}{4 -(s+\sigma)^2} \mathrm{d}s$$ at the pole $s=1-iT$ is $\ll T^{-2}$. How can this be proved ?
ADDENDUM: Maybe my question isn't explicitly clear, hence the votes for closure.
But i'm asking for the exact expression of the quantity (integral) that Titchmarsh claims to be $\ll T^{-2}$.