Gravitating lattice Fourier kernel ($F(\xi )=2{\sum }_{l\ge 1}(1-\cos (l\xi ))/l^3$)

The discrete normal modes of an infinite gravitating particle lattice involve $F(\xi )=2{\sum }_{l=1}^{\infty }(1-\cos (l\xi ))/l^3$. It is even, nonnegative and $2\pi$-periodic. Its derivative is a positive reciprocal-square sine series for $0<\xi <\pi$, so its maxima occur exactly at odd multiples of $\pi$, with $F(\pi )=(7/2)\zeta (3)$.