Integral of a logarithm containing exponentials

Question

I would like to know the exact expression for the following integral $$ I_{p,q}=\int_{0}^{2\pi}\int_{0}^{2\pi}e^{ip\alpha}e^{iq\beta}ln\left|sin\left(\frac{\alpha-\beta}{2}\right)\right|d\alpha d\beta $$ where $p$ and $q$ are integers. I can run individual integrals in Mathematica, but it won't do the general integral, even when I add the assumptions that $p$ and $q$ are integers. Some examples that the exact solution should be consistent with are: $$ I_{0,0}=-4\pi^{2}ln(2) $$ $$ I_{-1,1}=I_{1,-1}=-2\pi^{2} $$ $$ I_{-2,2}=I_{2,-2}=-\pi^{2} $$ and I think all integrals not of the form $I_{-q,q}$ vanish. Thanks in advance for any help.