  An asymptotic formula for $$ \int_{T/2}^{T}Z^2(t)Z(t+U)\,dt\qquad(0< U = U(T)
\le T^{1/2-\varepsilon}) $$ is derived, where $$ Z(t) :=
\zeta(1/2+it){\bigl(\chi(1/2+it)\bigr)}^{-1/2}\quad(t\in\Bbb R), \quad \zeta(s)
= \chi(s)\zeta(1-s) $$ is Hardy's function. The cubic moment of $Z(t)$ is also
discussed, and a mean value result is presented which supports the author's
conjecture that $$ \int_1^TZ^3(t)\,dt \;=\;O_\varepsilon(T^{3/4+\varepsilon}).
$$
