  Recently Witten conjectured the existence of a family of "extremal" conformal
field theories (ECFTs) of central charge c=24k, which are supposed to be dual
to three-dimensional pure quantum gravity in AdS3. Assuming their existence, we
determine explicitly the genus two partition functions of k=2 and k=3 ECFTs,
using modular invariance and the behavior of the partition function in
degenerating limits of the Riemann surface. The result passes highly nontrivial
tests and in particular provides a piece of evidence for the existence of the
k=3 ECFT. We also argue that the genus two partition function of ECFTs with
k<11 are uniquely fixed (if they exist).
