Euler product

(1)   \begin{equation*}\zeta(2) = \frac{\pi^2}{\sqrt{105}}\prod_{n=1}^\infty \left(1-\frac{2}{p_n^{2}+p_n^{-2}}\right)^{-1/2}\end{equation*}

(2)   \begin{equation*}\zeta(3) = \pi^3 \sqrt{\frac{691}{675675}}\prod_{n=1}^\infty \left(1-\frac{2}{p_n^{3}+p_n^{-3}}\right)^{-1/2}\nonumber\end{equation*}

(3)   \begin{equation*}\zeta(4) = \pi^4 \sqrt{\frac{3617}{34459425}}\prod_{n=1}^\infty \left(1-\frac{2}{p_n^{4}+p_n^{-4}}\right)^{-1/2}\nonumber\end{equation*}

(4)   \begin{equation*}\zeta(5) = \pi^5 \sqrt{\frac{174611}{16368226875}}\prod_{n=1}^\infty \left(1-\frac{2}{p_n^{5}+p_n^{-5}}\right)^{-1/2}\nonumber\end{equation*}

(5)   \begin{equation*}\zeta(6) = \pi^6 \sqrt{\frac{236364091}{218517792968475}}\prod_{n=1}^\infty \left(1-\frac{2}{p_n^{6}+p_n^{-6}}\right)^{-1/2}\nonumber\end{equation*}

(6)   \begin{equation*}\zeta(7) = \pi^7 \sqrt{\frac{3392780147}{30951416768146875}}\prod_{n=1}^\infty \left(1-\frac{2}{p_n^{7}+p_n^{-7}}\right)^{-1/2}\nonumber\end{equation*}

(7)   \begin{equation*}\zeta(8) = \pi^8 \sqrt{\frac{7709321041217}{694097901592400930625}}\prod_{n=1}^\infty \left(1-\frac{2}{p_n^{8}+p_n^{-8}}\right)^{-1/2}\nonumber\end{equation*}

(8)   \begin{equation*}\zeta(9) = \pi^9 \sqrt{\frac{26315271553053477373}{23383376494609715287281703125}}\prod_{n=1}^\infty \left(1-\frac{2}{p_n^{9}+p_n^{-9}}\right)^{-1/2}\nonumber\end{equation*}

(9)   \begin{equation*}\zeta(10) = \pi^{10} \sqrt{\frac{261082718496449122051}{2289686345687357378035370971875}}\prod_{n=1}^\infty \left(1-\frac{2}{p_n^{10}+p_n^{-10}}\right)^{-1/2}\nonumber\end{equation*}

(10)   \begin{equation*}\zeta(11) = \pi^{11} \sqrt{\frac{2530297234481911294093}{219012470258383844016431785453125}}\prod_{n=1}^\infty \left(1-\frac{2}{p_n^{11}+p_n^{-11}}\right)^{-1/2}\nonumber\end{equation*}