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Consider the infinite series $$ F(s)=\sum_{k=1}^\infty \frac{1}{k^{2s+1} \sin(k \pi \sqrt{2})} $$ Prudnikov Vol.1 , gives a result for this sum in 5.4.16.13 for $s=1.$ $$ F(1)=-\frac{13 \pi^3}{360 \...