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Is there a procedure/algorithm for calculating sums of the form

$$ \sum_{n_1,\ldots,n_r >0} \frac1{L_1(n_1,\ldots,n_r)^{m_1} \ldots L_r(n_1,\ldots,n_r)^{m_r}} $$ where $$ L_i(n_1,\ldots, n_r) := a_{1,i} n_1 + \ldots + a_{r,i} n_r $$ for some $a_{i,j} \in \mathbb C$. How would one go about evaluating this sum to say 20 or 30 digits?

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