Euler the magician

If $\chi$ is the nontrivial Dirichlet character modulo $4$ , so that $\chi(n) \; = \; \sin \tfrac{n\pi}{2} \; = \; \left\{ \begin{array}{lll} 1 & \hspace{1cm} & n \equiv 1 \pmod{4}\\ -1 & & n \equiv 3 \pmod{4} \\ 0 & & \mathrm{o.w.} \end{array} \right.$ then we are interested in the product $P_+ \; = \; \prod_{p > 2} \left(1 + \tfrac{\chi(p)}{p}\right)$ It is a well-known result that the product $P_- \; = \; \prod_{p > 2}\left(1 - \tfrac{\chi(p)}{p}\right)$ is such that $P_-^{-1} \; = \; \tfrac14\pi$ On the other hand $P_+P_- \; =\; \prod_{p > 2} \left(1 - \tfrac{1}{p^2}\right) \; = \; \tfrac43\prod_p \left(1 - \tfrac{1}{p^2}\right) \; = \; \tfrac43 \zeta(2)^{-1} \; = \; \tfrac{8}{\pi^2}$ and hence $P_+ \; = \; \tfrac{8}{\pi^2} \times \tfrac14\pi \; = \; \tfrac{2}{\pi} \; = \; \boxed{0.63661977}$