You need not any tables to this end. Mathematica does the job by

f[a_, b_] := NSum[1/(k*n*(a*k^2 + b*n^2)), {k, 1, Infinity}, {n, 1, Infinity}]

For example,

f[Pi,0.1]

$0.944319 $

I have stroung doubts concerning a closed-form expression for this sum.

You need not any tables to this end. Mathematica does the job by

f[a_, b_] := NSum[1/(k*n*(a*k^2 + b*n^2)), {k, 1, Infinity}, {n, 1, Infinity}]

For example,

f[Pi,0.1]

$0.944319 $

I have strong doubts concerning a closed-form expression for this sum.