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 $ζ(2m)$, $m\in\{0,\ldots,\lfloor j/2\rfloor\}$, of the Riemann zeta function $ζ$ 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.