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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 ...

Let $\lambda_n$ be an increasing and unbounded sequence of positive real numbers and $a_n$ be a sequence of real numbers such that $$\sum_{n=1}^\infty a_n \lambda_n^k=0 \ \ \text{ for all }\ \ k\geq ...