  Formulas are provided for the cumulants and the moments of the time $T$ back
to the most recent common ancestor of the Kingman coalescent. It is shown that
both the $j$th cumulant and the $j$th moment of $T$ are linear combinations of
the values $\zeta(2m)$, $m\in\{0,\ldots,\lfloor j/2\rfloor\}$, of the Riemann
zeta function $\zeta$ with integer coefficients. The proof is based on a
solution of a two-dimensional recursion with countably many initial values. A
closely related strong convergence result for the tree length $L_n$ of the
Kingman coalescent restricted to a sample of size $n$ is derived. The results
give reason to revisit the moments and central moments of the classical Gumbel
distribution.
