  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 $E(T)\ll_{\varepsilon}
T^{\frac{1}{3}+\varepsilon}$.
