On the estimation of $Z_2(s)$

Estimates for $Z_2(s)={\int }_1^{\mathrm{\mid }}infty\mathrm{\mid }\zeta (1\mathrm{/}2+ix){\mathrm{\mid }}^4{x}^{-s}dx(\mathrm{\Re }s>1)$ are discussed, both pointwise and in mean square. It is shown how these estimates can be used to bound $E_2(T)$, the error term in the asymptotic formula for ${\int }_0^T\mathrm{\mid }\zeta (1\mathrm{/}2+it){\mathrm{\mid }}^{4}dt$.