  In this paper we study two functions $F(x)$ and $J(x)$, originally found by
Herglotz in 1923 and later rediscovered and used by one of the authors in
connection with the Kronecker limit formula for real quadratic fields. We
discuss many interesting properties of these functions, including special
values at rational or quadratic irrational arguments as rational linear
combinations of dilogarithms and products of logarithms, functional equations
coming from Hecke operators, and connections with Stark's conjecture. We also
discuss connections with 1-cocycles for the modular group
$\mathrm{PSL}(2,\mathbb{Z})$.
