The Fourier transform of a function of $\ln(\sin(x))$

Question

In order to make the Fourier transform of the function $f\left(l\right) =\ln \sin\left(\frac{\pi l}{N}\right)$, we shall have $$ F(q) = \frac{1}{N}\sum_{l=0}^{N} e^{-iql}f(l) $$ where $f(0)=f(N)=0$ is defined. So, we want to calculate the summation $$ F(q) = \frac{1}{N}\sum_{l=1}^{N-1} e^{-iql}\ln \sin\left(\frac{\pi l}{N}\right). $$ What is the closed-form?