  Shanks conjectured that $\zeta ' (\rho)$, where $\rho$ ranges over
non-trivial zeros of the Riemann zeta function, is real and positive in the
mean. We present a history of this problem, including a generalisation to all
higher-order derivatives $\zeta^{(n)}(s)$, for which the sign of the mean
alternatives between positive for odd $n$ and negative for even $n$.
Furthermore, we give a simple heuristic that provides the leading term
(including its sign) of the asymptotic formula for the average value of
$\zeta^{(n)}(\rho)$.
