  Some new results on power moments of the integral $$ J_k(t,G) =
{1\over\sqrt{\pi}G} \int_{-\infty}^\infty |\zeta(1/2 + it + iu)|^{2k}{\rm
e}^{-(u/G)^2}du \qquad(t \asymp T, T^\epsilon \le G \ll T, k\in\N) $$ are
obtained when $k=1$. These results can be used to derive bounds for moments of
$|\zeta(1/2+it)|$.
