On the moments of Hecke series at central points II

We prove, in standard notation from spectral theory, the asymptotic formula ($B>0$) $$\sum _{{\kappa }_j\le T}{\alpha }_j{H}_j(1\mathrm{/}2)=(\frac{T}{\pi }{)}^2-BT\log T+O(T(\log T{)}^{1\mathrm{/}2}),$$ by using an approximate functional equation for $H_j(1\mathrm{/}2)$ and the Bruggeman–Kuznetsov trace formula. We indicate how the error term may be improved to $O(T(\log T{)}^{ϵ})$.