Kontsevich-Zagier's article "Periods" contains the following question

Is $\displaystyle \sum_{x,y,z \in \mathbb{Z}}' \frac{1}{(x^2+y^2+z^2)^2}$ an extended period?

($\sum'$ means we do not sum over $(x,y,z) = (0,0,0)$.)

Here period is loosely interpreted as "integral of algebraic differential forms (on an smooth projective algebraic variety) over algebraic subvariety". Extended period is a period up to a factor of integer powers of $\pi$.

(I am intentionally making this vague since I don't think even Kontsevich-Zagier tried to pin this down very precisely, but hopefully you know a period when you see one.)

**My question:** Is this now known? (It has been 15 years since Kontsevich-Zagier first published the article) More generally in this setup, the question would be

Let $Q(x_1,\cdots,x_n)$ be a positive definite quadratic form in an odd number ($\ge 3)$ of variables with coefficients in $\mathbb{Q}$. Is $\displaystyle \sum_{x_1,\cdots,x_n \in \mathbb{Z}} \frac{1}{Q(x_1,\cdots,x_n)^s}$ an extended period for $s > n/2$?

**Some background**
There is a beautiful theorem due to Beilinson and Deninger-Scholl,

TheoremLet $f$ be a modular form of weight $k \ge 2$ defined over $\overline{\mathbb{Q}}$. Then $L(f,m)$ are extended periods for all integers $m \ge k$.

(When $m < k$ this is still true; here I am merely distinguishing $m$ by whether it is a critical value or not for $L(f,s)$.)

The question above for quadratic forms with even number of variables is then solved by this theorem, since it is the $L$-function for the theta series $\sum'_{x_1,\cdots,x_n} q^{Q(x_1,\cdots,x_n)}$ of weight $n/2$ if $n$ is even.

I tried to look up generalization of this theorem to half-integral weight modular forms with no success.