Let $\Lambda_8$ be the $E_8$ lattice.
The functional equation is:
$$f(a) = \sum\limits_{a=b+c} f(b)f(c)$$
Where $f(a)$ is defined only for points on the lattice $a\in \Lambda_8$
What would be a general solution? I can find at least one solution: $f(0)=1$, $f(a)=0$ for $a\neq 0$