  We formulate a generalization of a `refined class number formula' of Darmon.
Our conjecture deals with Stickelberger-type elements formed from generalized
Stark units, and has two parts: the `order of vanishing' and the `leading
term'. Using the theory of Kolyvagin systems we prove a large part of this
conjecture when the order of vanishing of the corresponding complex
$L$-function is $1$.
