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We compute the average of a product of two shifted squares of the Riemannzeta on the critical line with shifts up to size $T^{3/2-\varepsilon}$. We givean explicit expression for such an average and derive an approximate spectralexpansion for the error term similar to Motohashi's. As a consequence, we alsocompute the $(2,2)$-moment of moment of the Riemann zeta, for which wepartially verify (and partially refute) a conjecture of Bailey and Keating.

Correlations of the squares of the Riemann zeta on the critical line