The central value of the triple sine function (Q1017357)

In the paper, the investigations on the triple sine function \[ S_3(x, (\omega_1, \omega_2, \omega_3))=\mathop{{\prod}^*}\limits_{n_1, n_2, n_3\geq 0} (n_1\omega_1+n_2\omega_2+n_3\omega_3+x)\times \mathop{{\prod}^*}\limits_{m_1, m_2, m_3\geq 1} (m_1\omega_1+m_2\omega_2+m_3\omega_3-x), \] where \(\mathop{{\prod}^*}\limits_\lambda =\exp\left(-{\partial\over\partial s}\sum\limits_\lambda \lambda^{-s}|_{s=0}\right)\), introduced and studied by \textit{the author} and \textit{S. Koyama} [Proc. Japan Acad., Ser. A 68, No. 9, 256--260 (1992; Zbl 0797.11053), Forum Math. 15, No. 6, 839--876 (2003; Zbl 1065.11065)], are continued. The author gives the explicit integral expression for the central value of \(S_3(x, (\omega_1, \omega_2, \omega_3))\). More precisely, he proves that, for \(\omega_1, \omega_2, \omega_3 >0\), \[ \begin{multlined} S_3\left({\omega_1+\omega_2+\omega_3\over 2}, (\omega_1, \omega_2, \omega_3)\right)\\ = \exp\left(-\int_0^\infty\left({1\over 4}\prod_{k=1}^3\left(\sinh\left({\sqrt{2}\omega_kt\over \sqrt{\omega_1^2+\omega_2^2+\omega_3^2}}\right)\right)^{-1} -{(\omega_1^2+\omega_2^2+\omega_3^2)^{3/2}\over 8\sqrt{2}\omega_1\omega_2\omega_3t^3}\left(1-{t^2\over 3}\right)\right){d t\over t}\right). \end{multlined} \] Moreover, he observes that the left-hand side of this equality lies in the interval \((0, 1)\), and presents the graph of \(S_3(x, (\omega_1, \omega_2, \omega_3))\) in the fundamental domain \(0\leq x\leq \omega_1+\omega_2+\omega_3\). As an application, the expression for \(\zeta(3)\) is presented, i.e. the following equality \[ \zeta(3)={16\pi ^2\over 3}\int_0^\infty\left(2(e^{\sqrt{2/3}t}-e^{-\sqrt{2/3}t})^{-3}+{3\over 16}\sqrt{2\over 3}\left({1\over t}-{3\over t^3}\right)\right){d t\over t}-{2\over 3}\pi^2\log 2 \] is obtained.