Let $$I(T)=\int_0^T \left|\zeta\left(\frac{1}{2} + i t\right)\right|^2 dt$$ and let $E(T)$ be $I(T)$ minus what turn out to be its main terms: $$E(T) = I(T)- T \log \frac{T}{2 \pi} - (2 \gamma - 1) T.$$
Atkinson (1949) gave an explicit formula for $E(T)$. As it happens, his formula (see the original paper, or Chapter 15 of Ivić's book) has a small error term (namely, $O(\log^2 T)$) whose constant is not made explicit. Working the constant out looks like something that would involve plenty of time and ink, but should be doable.
Has anybody done it?
(And what about the case of the integral over a line with $\Re s=\sigma$, $0<\sigma<1$, $\sigma$ other than $1/2$? I know Matsumoto and Meurman have a version of Atkinson's formula in that case, but they also have a small, non-explicit error term, similar to the above.)
