The Laplace and Mellin transforms of powers of the Riemann zeta-function

This paper gives a survey of known results concerning the Laplace transform $$L_k(s):={\int }_0^{\mathrm{\infty }}\mathrm{\mid }\zeta (1\mathrm{/}2+ix){\mathrm{\mid }}^{2k}{\mathrm{e}}^{-sx}\mathrm{d}x(k\in N,\mathbb{R}s>0),$$ and the (modified) Mellin transform $${\mathcal{Z}}_k(s):={\int }_1^{\mathrm{\infty }}\mathrm{\mid }\zeta (1\mathrm{/}2+ix){\mathrm{\mid }}^{2k}{x}^{-s}\mathrm{d}x(k\in N),$$ where the integral is absolutely convergent for $\mathbb{R}s\ge c(k)>1$. Also some new results on these integral transforms of $\mathrm{\mid }\zeta (1\mathrm{/}2+ix){\mathrm{\mid }}^{2k}$ are given, which have important connections with power moments of the Riemann zeta-function $\zeta (s)$.