In my research, I came across a 1-cocycle in the following group cohomology complex:

Let $\Lambda_\mathbb{Z}$ be a lattice (i.e. isomorphic to $\mathbb{Z}^n)$; let $\Lambda_\mathbb{C} = \Lambda_\mathbb{Z} \otimes_\mathbb{Z} \mathbb{C}$; let $\mathbb{C}(\Lambda_\mathbb{C})$ be the rational functions on $\Lambda_\mathbb{C}$, and let the $\times$ denote the multiplicative group.

Let $\Lambda_\mathbb{Z}$ act on $\mathbb{C}(\Lambda_\mathbb{C})$ by translation. Then the bar resolution takes the form $\mathbb{C}(\Lambda_\mathbb{C})^\times \rightarrow Fun(\Lambda_\mathbb{Z}, \mathbb{C}(\Lambda_\mathbb{C})^\times) \rightarrow Fun(\Lambda_\mathbb{Z}^2, \mathbb{C}(\Lambda_\mathbb{C})^\times) \rightarrow \ldots$

where $Fun$ refers to arbitrary functions.

Is anything known about the cohomology of this complex? I think I've figured out the first cohomology group (the group is not easy to explain, so I'd rather not type it out if there is an immediate answer), and the zeroth cohomology group is clearly $\mathbb{C}$ (as the 0-cocycles are constant functions).

For a more general question, is anything known about cohomology with coefficients in the multiplicative group of a field?