I find a reference see Chavan and Vignat - Evaluation of multivariate integrals based on a duality identity for the Stieltjes transform we have :
$$\int_{0}^{\frac{\pi}{2}}\int_{0}^{\frac{\pi}{2}}\frac{\ln\left(\cos\left(\frac{x}{2}\right)\right)-\ln\left(\cos\left(\frac{z}{2}\right)\right)}{\cos\left(x\right)-\cos\left(z\right)}dxdz=\pi G-\frac{7}{4}\zeta(3).$$
Then we can use $a,x\in[0,\pi/2)$ :
$$f\left(x\right)=\frac{\ln\left(\cos\left(\frac{x}{2}\right)\right)-\ln\left(\cos\left(\frac{z}{2}\right)\right)}{\cos\left(x\right)-\cos\left(z\right)},g\left(x\right)=\frac{e^{\frac{x}{4}}}{2}+\frac{e^{-\frac{x}{4}}}{2}-1+f\left(0\right),h\left(x\right)=f\left(x\right)-g\left(x\right).$$
Then we have : \begin{gather*} f(x)= g\left(x\right)+\frac{h''\left(0\right)}{2}x^{2}+\frac{h''''\left(0\right)}{4!}x^{4}+\dotsb \\ \int_{0}^{\frac{\pi}{2}}\int_{0}^{\frac{\pi}{2}}\left(g\left(x\right)+\frac{h''\left(0\right)}{2}x^{2}+\frac{h''''\left(0\right)}{4!}x^{4}+\frac{h''''''\left(0\right)}{6!}x^{6}+\dotsb\right)dxdz.\tag{I}\label{I} \end{gather*}
My motivation is related to Show that : $\frac{1}{\zeta(3)}<2C-1$ . In fact I tried to find a sufficently accurate series to evaluate the product in question .
Question :
How does \eqref{I} look in terms of values or how to get the series equal to the integral in terms of known value instead of valued function ?