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Let F be a non-archimedian local field of characteristic 0, and O the ring ofintegres in F. We give an explicit formula for the Siegel series of ahalf-integral matrix over O. This formula expresses the Siegel series of ahalf-integral matrix $B$ explicitly in terms of the Gross-Keating invariant of$B$ and its related invariants.

An explicit formula for the Siegel series of a quadratic form over a non-archimedian local field