  The coefficient of the $D^6 {\cal R}^4$ interaction in the low energy
expansion of the two-loop four-graviton amplitude in type II superstring theory
is known to be proportional to the integral of the Zhang-Kawazumi (ZK)
invariant over the moduli space of genus-two Riemann surfaces. We demonstrate
that the ZK invariant is an eigenfunction with eigenvalue 5 of the
Laplace-Beltrami operator in the interior of moduli space. Exploiting this
result, we evaluate the integral of the ZK invariant explicitly, finding
agreement with the value of the two-loop $D^6 {\cal R}^4$ interaction predicted
on the basis of S-duality and supersymmetry. A review of the current
understanding of the $D^{2p} {\cal R}^4$ interactions in type II superstring
theory compactified on a torus $T^d$ with $p \leq 3$ and $d \leq 4$ is
included.
