  We give explicit formulas for the Kawazumi-Zhang invariant and Faltings
delta-invariant of a compact and connected Riemann surface of genus three. The
formulas are in terms of two integrals over the associated jacobian, one
integral involving the standard Riemann theta function, and another involving a
theta function particular to genus three that was discovered by Frobenius. We
review part of Frobenius' work on his theta function and connect our results
with a formula due to Bloch, Hain and Bost describing the archimedean height
pairing of Ceresa cycles in genus three.
