Formula for designing Lynn's low pass filter
Just analyzing your equation, let $a = f_c / f_s$
Also we know that for $m = k/(2a)$ it reduces to $1/(m^2 sin(\pi a))$ for odd $k$ and $0$ for even $k$.
$$\lim_{m \to 0} \frac{sin^2( \pi m a)}{m^2 sin(\pi a)} = \frac{a^2 \pi^2}{sin(\pi a)} = \frac{1}{\sqrt{2}}$$
Even for this case, determining $a$ is not an easy task. I computed numerically the curve defined by this equation.