  We prove a unicity result for the $L$-functions appearing in the
non-commutative Iwasawa main conjecture over totally real fields. We then
consider continuous representations $\rho$ of the absolute Galois group of a
totally real field $F$ on adic rings in the sense of Fukaya and Kato. Using our
unicity result, we show that there exists a unique sensible definition of a
non-commutative $L$-function for any such $\rho$ that factors through the
Galois group of a possibly infinite totally real extension. We also consider
the case of CM-extensions and discuss the relation with the equivariant main
conjecture for realisations of abstract $1$-motives of Greither and Popescu.
