  We prove an analogue of Kronecker's second limit formula for a continuous
family of "indefinite zeta functions". Indefinite zeta functions were
introduced in the author's previous paper as Mellin transforms of indefinite
theta functions, as defined by Zwegers. Our formula is valid in dimension g=2
at s=1 or s=0. For a choice of parameters obeying a certain symmetry, an
indefinite zeta function is a differenced ray class zeta function of a real
quadratic field, and its special value at $s=0$ was conjectured by Stark to be
a logarithm of an algebraic unit. Our formula also permits practical
high-precision computation of Stark ray class invariants.
