On the density of unnormalized Tamagawa numbers of orthogonal groups, II

This is the second part of a series of three papers. In this series of papers, we determine the density of unnormalized Tamagawa numbers of projective special orthogonal groups defined over a fixed number field. In this part, we determine local constants at infinite places which appear in the density theorem and prove the density theorem when the number of variables $n$ is odd. For that purpose we make a Dirichlet series whose coefficients are unnormalized Tamagawa numbers of projective special orthogonal groups and prove that it is rather simple when $n$ is odd.