Jacob's ladders and the asymptotic formula for the integral of the eight order expression $\mathrm{\mid }\zeta (1\mathrm{/}2+i{\text{vp}}_2(t)){\mathrm{\mid }}^4\mathrm{\mid }\zeta (1\mathrm{/}2+it){\mathrm{\mid }}^4$

It is proved in this paper that there is a fine correlation between the values of $\mathrm{\mid }\zeta (1\mathrm{/}2+i{\text{vp}}_2(t)){\mathrm{\mid }}^4$ and $\mathrm{\mid }\zeta (1\mathrm{/}2+it){\mathrm{\mid }}^4$ where ${\text{vp}}_2(t)$ stands for the Jacob’s ladder of the second order. This new asymptotic formula cannot be obtained in known theories of Balasubramanian, Heath-Brown and Ivic.