Let $-D < 0$ be a negative fundamental discriminant and let $y$ range over the values $y = y_Q = \frac{\sqrt{|D|}}{2a}$, as the values $(a,b,c)$ run through the reduced binary quadratic forms $Q = aX^2+bXY+cY^2$ of discriminant $b^2-4ac = -D$ (thus $\mathrm{gcd}(a,b,c) = 1$, $-a < b \leq a \leq c$). Those are the ordinates of the CM points $z_Q = \frac{b + \sqrt{-D}}{2a}$ in the standard fundamental domain for $\mathcal{H} / \mathrm{PSL}(2,\mathbb{Z})$. By Duke's theorem, the $z_Q$ are equidistributed in the measure $\frac{3}{\pi} \frac{dx \, dy}{y^2}$. The highest lying point has ordinate $k := \sqrt{|D|}/2$, and since $\int_1^k t \frac{dt}{t^2} = \log{k}$, a naive heuristic suggests $\log{k}$ for the expected mean value of $\frac{\pi}{3}y_Q$. This turns out to be correct assuming the Riemann hypothesis for $L(s,\chi_D)$, as can be proved departing from the limit formula of Kronecker, Chowla and Selberg.

Question. Assume GRH, and possibly some other standard analytic hypotheses. What can be said about the mean value of $\frac{\pi}{3} y_Q \log{y_Q}$, asymptotically as $D \to \infty$?

The naive heuristic would suggest here a main term of $\frac{1}{2} (\log{k})^2$. Is this any close to the truth?

Of course, the same question could be asked for the positive discriminants (indefinite binary quadratic forms and closed geodesics on the modular surface).