Refined class number formulas for ${\mathbb{G}}_m$

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 $1$.