  We consider the sum $\sum 1/\gamma$, where $\gamma$ ranges over the ordinates
of nontrivial zeros of the Riemann zeta-function in an interval $(0,T]$, and
consider the behaviour of the sum as $T \to\infty$. We show that, after
subtracting a smooth approximation $\frac{1}{4\pi} \log^2(T/2\pi),$ the sum
tends to a limit $H \approx -0.0171594$ which can be expressed as an integral.
We calculate $H$ to high accuracy, using a method which has error $O((\log
T)/T^2)$. Our results improve on earlier results by Hassani and other authors.
