  Some results and conjectures on $Z_2(s) = \int_1^\infty
|\zeta(1/2+ix)|^4x^{-s}dx (\Re s > 1)$ are presented. Consequences of these
conjectures regarding the eighth moment of $|\zeta(1/2+it)$ and the error term
in the fourth moment of $|\zeta(1/2+it)$ are discussed.
