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?