Atkinson's formula for the mean square of $\zeta (s)$ with an explicit error term

We provide an explicit $O\left({\log }^{2}T\right)$-term of the celebrated Atkinson’s formula for the error term $E(T)$ of the second power moment of the Riemann zeta-function on the critical line. As an application, we obtain an explicit version of well-known estimate $E(T)\ll_{\varepsilon} T^{\frac{1}{3}+\varepsilon}$.