Let \(\zeta^* (s,x) = \zeta (s,x + 1)\), where \(\zeta (s,a)\) \((0 < a \leq 1)\) is the Hurwitz zeta-function. The author proves that \[ \int^ 1_ 0 \left | \zeta^* \Bigl( {1 \over 2} + it, x \Bigr) \right |^ 2 dx = \log \left( {t \over 2 \pi} \right) + \gamma - 2 \text{Re} {\zeta (1/2 + it) \over 1/2 + it} + O \left( {1 \over t} \right), \] where \(\zeta (s)\) is the Riemann zeta-function and \(\gamma\) is Euler's constant. \textit{M. Katsurada} and \textit{K. Matsumoto} [Proc. Japan Acad., Ser. A 69, 303- 307 (1993; Zbl 0799.11027)] proved a general mean value formula for \(\zeta (s,a)\), which in particular implies a full asymptotic expansion of the integral in (1), thereby sharpening the author's result.