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.
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