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It is shown in this paper that there is a fine correlation of the fifth orderbetween the values $Z[\phi(t)/2+\rho_1]Z[\phi(t)/2+\rho_2]Z[\phi(t)/2+\rho_3]$and $\hat{Z}^2(t)$ which correspond to two collections of disconnected sets.This new asymptotic formula cannot be obtained within known theories ofBalasubramanian, Heath-Brown and Ivic.

Jacob's ladders and the first asymptotic formula for the expression of  the fifth order $Z^3[...]\hat{Z}^2(t)$ for the collection of disconnected  sets