We introduce a smoothed version of the equivariant
$L$-function for one-dimensional admissible
extensions of number fields. Integrality of this smoothed
$L$-function, conjectured by Greenberg, has been verified for pro-
extensions (assuming the Equivariant Iwasawa Main Conjecture) as well as
$p$-abelian extensions (unconditionally). Integrality in the general case
is also expected to hold, and is the subject of ongoing research.