Sarnak and Strömbergsson studied Epstein zeta function $\zeta(L,s)=\sum\limits_{0\neq v\in L}\langle v,v\rangle^{-s}$ of a number of highly symmetric lattices in their Inventiones Math. paper. In particular, they show (Thm 1 in [loc.cit.]) that in dimensions $n=4,8,24$ one has $\zeta(L,s)$, with $s>0$, reaching a strict local minimum $\zeta(L_n,s)$ on $L_4=D_4,$ $L_8=E_8$ and the Leech lattice $L_{24}=\Lambda_{24}$, respectively.
Their analysis has two parts - first they need to show that $\zeta(L_n,s)<0$ for $\frac{n}{2}>s>0$, and $L_n$ as above. Then they study polynomial invariants of the automorphism groups of $L_n$ (acting on bilinear forms), and establish that these only start to differ from the corresponding invariants of the full orthogonal group $O(n)$ at high enough degrees, allowing for an analytic argument to complete the proof.
It turns out that for Barnes-Wall lattice $\Lambda_{16}$ the needed property of the invariant ring holds, too - so their Thm 1 could apparently be extended to the case $n=16$ if $\zeta(\Lambda_{16},s)<0$ for $8>s>0$ holds.
So my question is whether $\zeta(\Lambda_{16},s)$ can be found in the literature. Note that for $n=4,8,24$ the authors were able to use information in Conway-Sloane book (I'm struggling to fill in details there, though) - but the info on $\Lambda_{16}$ there appears to be less complete.
