Let's say I have a motive in $\mathcal{M}_{num}(K)$ ($K$ a number field). For each prime $l$ there is a realization of this motive in terms of etale cohomology with coefficients in $\mathbb{Q}_l$. This has more structure than being a mere vector space: it is a representation of $Gal(K)$! This representation has an $L$-function. As I understand it, the $L$ function doesn't depend on $l$.

What I wonder is whether there is something special about etale cohomology with coefficients in an $l$-adic field, or whether every Weil cohomology has an $L$-function attached to it that would equal the $L$-function of any other realization.

The usual (singular) cohomology with the complex topology, is also a representation of $Gal(K)$ (factoring through the motivic Galois group, if this means anything to you). Is it true, then, that the $L$-function attached to this representation is the same as the $L$-function of the realizations via etale cohomology with coefficients in $\mathbb{Q}_l$?

How does one go about proving equalities between $L$-functions of different realizations of the same motive?