Sarnak and Strömbergsson studied [Epstein zeta function](https://encyclopediaofmath.org/wiki/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](http://www2.math.uu.se/%7Eastrombe/papers/minima.pdf).
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](https://link.springer.com/book/10.1007/978-1-4757-6568-7) book (I'm struggling to fill in details there, though) - but the info on $\Lambda\_{16}$ there appears to be less complete.



Epstein zeta function of Barnes-Wall and related lattices
It turns out this question, and more, is basically answered, for all Barnes-Wall lattices $\Lambda\_{n}$, in affirmative in [Spherical designs and zeta functions of lattices](https://arxiv.org/abs/math/0611735) - the published version [here](https://doi.org/10.1155/IMRN/2006/49620) with the extra assumption $s>n/2$.


It's not clear from this text whether for $n=16$ the assumption $s>8$ is needed, but it's remarked that it's needed for $n$ big enough.


