Let $a=\pi h,$ $h$ positive integer and $$J_h = \int_{0}^{\alpha} \ln\Bigg( \frac{y^{h}\sin{y}}{\prod_{j=1}^{h} (\pi j-y)}\Bigg) dy$$ The function $$f_h (y)= \frac{y^{h}\sin{y}}{\prod_{j=1}^{h} (\pi j-y)}$$ is positive in $(0,\alpha)-\{\pi,2\pi,...,(h-1)\pi\}.$ $\ln{f_h(y)}$ is not defined in $\{j\pi:j=0,1,...,h\}$ but can be continued extended for $j=1,2,...,h$ (the limits $\lim_{y\rightarrow \pi j}\ln(f_h(y))$ exist for $j=1,2,...,h$ but not for $j=0$). Although the integral exists, from Riemann - Lebesgue Theorem.
Some values are (computed with computer), $$J_1=-\pi \ln{2}, J_2 = 0, J_3 = 6\pi \ln(3) - 7\pi\ln(2),$$ the code can be found https://sagecell.sagemath.org/?q=gmcnde. If someone tries to "break" the $\ln$ , then the $\ln$ of the denominator is not well defined in some subintervals of $(0,\alpha)$. Although for $h=1$ the calculation of $J_1$ is easy. Any idea on how to compute it for $h>1?$
This is not an answer but it is too long for comments.
Apparently $$J_h=\pi \log \left(\frac{p_h}{q_h}\right)$$
Now, observe the prime factorizations for the $p_h$ and $q_h$
$$\left( \begin{array}{ccc} h & \text{factors of } p_h & \text{factors of } q_h \\ 3 & \left( \begin{array}{cc} 3 & 6 \end{array} \right) & \left( \begin{array}{cc} 2 & 7 \end{array} \right) \\ 4 & \left( \begin{array}{cc} 2 & 16 \end{array} \right) & \left( \begin{array}{cc} 3 & 6 \end{array} \right) \\ 5 & \left( \begin{array}{cc} 5 & 20 \end{array} \right) & \left( \begin{array}{cc} 2 & 25 \\ 3 & 6 \end{array} \right) \\ 6 & \left( \begin{array}{cc} 2 & 4 \\ 3 & 24 \end{array} \right) & \left( \begin{array}{cc} 5 & 10 \end{array} \right) \\ 7 & \left( \begin{array}{cc} 7 & 42 \end{array} \right) & \left( \begin{array}{cc} 2 & 39 \\ 3 & 18 \\ 5 & 10 \end{array} \right) \\ 8 & \left( \begin{array}{cc} 2 & 128 \end{array} \right) & \left( \begin{array}{cc} 3 & 18 \\ 5 & 10 \\ 7 & 14 \end{array} \right) \\ 9 & \left( \begin{array}{cc} 3 & 126 \end{array} \right) & \left( \begin{array}{cc} 2 & 89 \\ 5 & 10 \\ 7 & 14 \end{array} \right) \\ 10 & \left( \begin{array}{cc} 5 & 80 \end{array} \right) & \left( \begin{array}{cc} 3 & 54 \\ 7 & 14 \end{array} \right) \\ 11 & \left( \begin{array}{cc} 11 & 110 \end{array} \right) & \left( \begin{array}{cc} 2 & 111 \\ 3 & 54 \\ 5 & 30 \\ 7 & 14 \end{array} \right) \\ 12 & \left( \begin{array}{cc} 2 & 152 \\ 3 & 78 \end{array} \right) & \left( \begin{array}{cc} 5 & 30 \\ 7 & 14 \\ 11 & 22 \end{array} \right) \end{array} \right)$$
May be, this could give you some ideas.