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In this paper we obtain new complete hybrid formula for corresponding classof $\zeta$-factorization formulas. We demonstrate that this formula is thesynergetic one. Namely, this one describes the cooperation between some classof elementary functions and Riemann's zeta-function on a class of disconnectedsets on the critical line. Some comparizon between this pure mathematicalphenomenon and the Belousov-Zhabotinski chemical reaction is added.  \centering DEDICATED TO 110th ANNIVERSARY OF HARDY'S PURE MATHEMATICS

Jacob's ladders, crossbreeding, secondary crossbreeding and synergetic  phenomena generated by Riemann's zeta-function and some elementary functions  on disconnected sets of the critical line