  Thanks to Atkinson (1938), 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. Following both his work and the approach of Titchmarsh
(1986), we present an explicit version of the Atkinson formula, improving on a
recent bound by Simoni\v{c} (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).
