  Thanks to Littlewood (1922) and Ingham (1928), we know the first two terms of
the asymptotic formula for the square mean integral value of the Riemann zeta
function $\zeta$ on the critical line. Later, Atkinson (1939) presented this
formula with an error term of order $O(\sqrt{T}\log^{2}(T))$, which we call the
Atkinson formula. Following the latter approach and the work of Titchmarsh
(1986), we present an explicit version of the Atkinson formula, improving on a
recent bound by Simoni\v{c} (2020). Moreover, we extend the Atkinson formula to
the range $\Re(s)\in\left[\frac{1}{4},\frac{3}{4}\right]$, giving an explicit
bound for the square mean integral value of $\zeta$ and improving on a bound by
Helfgott and the authors (2019). We use mostly classical tools, such as the
approximate functional equation and the explicit convexity bounds of the zeta
function given by Backlund (1918).
