On the estimation of some Mellin transforms connected with the fourth moment of $\mathrm{\mid }\zeta (1\mathrm{/}2+it)\mathrm{\mid }$

Mean square estimates for $Z_2(s) = \int_1^\infty |\zeta(1/2+ix)|^4x^{-s}dx (\Re s > 1)$ are discussed, and some related Mellin transforms of quantities connected with the fourth power moment of $\mathrm{\mid }\zeta (1\mathrm{/}2+ix){\mathrm{\mid }}^4$.