Let $X$ be a scheme, $K^{\bullet}$ and $P^{\bullet}$ bounded complexes of abelian sheaves on $X\_{\rm ét}$.


I want to compute the hypercohomology:


$$\mathbb{H}^\*(X\_{\rm ét}, K^{\bullet}\otimes^L\_{\mathbf{Z}}P^{\bullet})$$


Is there a spectral sequence relating this to the étale hypercohomologies of $K^{\bullet}$ and $P^{\bullet}$, respectively?


Any references?



Spectral sequence for tensor product of complexes
I think that this situation is not quite suitable for spectral sequences. Having such a spectral sequence would be related to some compatibility of (derived) global sections and tensor product. In general there is a map 
$R\Gamma(X,A) \otimes^L R\Gamma(X,B) \to R\Gamma(X, A \otimes^L B)$ but I don't think that it is even close to be an isomorphism in general. Think for example on representations of $\mathbb{Z}/2$ in $\mathbb{F}\_2$ vector spaces (You can think of it as a sheaf of modules for a sheaf of rings over $spec(\mathbb{R})\_{et}$ is you like) and the modules $K = P = \mathbb{F}\_2$. Then the tensor product of the global sections in the derived sense and the sections of the tensor product are completely different. 


