On some conjectures and results for the Riemann zeta-function and Hecke series

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 $\mathrm{\mid }\zeta (1\mathrm{/}2+it)$ and the error term in the fourth moment of $\mathrm{\mid }\zeta (1\mathrm{/}2+it)$ are discussed.