Let G be a connected reductive quasi-split algebraic group defined over a p-adic field F of characteristic zero. For \pi irreducible admissible generic tempered representation of a standard Levi subgroup M of G, we prove that the normalized intertwining operators are holomorphic and nonvanishing on a set larger than the closure of the positive chamber of M, under some assumptions. As an application, we prove that if G is a split special orthogonal group (if G is even orthogonal, F can be archimedean) and \pi is an irreducible unitary representation of the Siegel Levi subgroup M of G, then the normalized intertwining operators are holomorphic and nonvanishing on a set larger than the closure of the positive chamber of M.