We introduce a smoothed version of the equivariant $S$-truncated $p$-adic Artin $L$-function for one-dimensional admissible $p$-adic Lie extensions of number fields. Integrality of this smoothed $p$-adic $L$-function, conjectured by Greenberg, has been verified for pro-$p$ 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.