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We firstly discuss properties of the L-group for Brylinski-Deligne (BD)extensions constructed by M. Weissman. Secondly, the Gindikin-Karpelevich (GK)formula for arbitrary BD extensions is computed and expressed in terms ofnaturally defined elements of the group. We show that the GK formula can beinterpreted as Langlands-Shahidi type L-functions associated with the ajointaction of the L-group of the Levi on certain Lie algebras. As a consequence,the constant term of Eisenstein series for BD extensions could be expressed interms of global (partial) Langlands-Shahidi type L-functions. In the end, weuse this to determine the residual spectrum of BD covers of some semisimplerank one groups.

The Gindikin-Karpelevich Formula and Constant Terms of Eisenstein Series for Brylinski-Deligne Extensions