On the estimation of some Mellin transforms connected with the fourth   moment of $|\zeta(1/2+it)|$
  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 $|\zeta(1/2+ix)|^4$. 
Aleksandar Ivi\'c