TITLE: What can be said about this double sum?
QUESTION [16 upvotes]: Question. Can this number be expressed in terms of classical values?
  $$\sum_{n,m=1}^{\infty}\frac1{(n^2+m^2)^{\frac32}}=1.056348517615643291\dots$$

UPDATE. I'm encouraged by Noam, Kevin and Igor's directional replies. To spice things up, I ask: is this true?
$$\sum_{n,m=1}^{\infty}\frac1{(n^2+m^2)^s}=\zeta(s)\beta(s)-\zeta(2s),$$
wherever convergence occurs. Here, $\zeta(s)$ and $\beta(s)$ are the Riemann zeta and Dirichlet beta functions, respectively.

REPLY [6 votes]: Exhaustive analysis of lattice sums can be found in the book Lattice sums then and now. This book can be found for free on the web.
OP's sum is given by formula 1.3.14 on page 33:
L. Lorenz. Bidrag tiltalienes theori. Tidsskrift Math., 1:97–114, 1871.