Let F be a number field and \(\pi \) a regular algebraic cuspidal automorphic representation of \(\textrm{GL}_N(\mathbb {A}_F)\) of symplectic type. When \(\pi \) is spherical at all primes \(\mathfrak {p}|p\) , we construct a p-adic L-function attached to any regular non-critical spin p-refinement \(\tilde{\pi }\) of \(\pi \) to Q-parahoric level, where Q is the (n, n)-parabolic. More precisely, we construct a distribution \(L_p(\tilde{\pi })\) on the Galois group \(\textrm{Gal}_p\) of the maximal abelian extension of F unramified outside \(p\infty \) and show that it interpolates all the standard critical L-values of \(\pi \) at p (including, for example, cyclotomic and anticyclotomic variation when F is imaginary quadratic). We show that \(L_p(\tilde{\pi })\) satisfies a natural growth condition; in particular, when \(\tilde{\pi }\) is ordinary, \(L_p(\tilde{\pi })\) is a (bounded) measure on \(\textrm{Gal}_p\) . As a corollary, when \(\pi \) is unitary, has very regular weight, and is Q-ordinary at all \(\mathfrak {p}|p\) , we deduce non-vanishing \(L(\pi \times (\chi \circ N_{F/\mathbb {Q}}),1/2) \ne 0\) of the twisted central value for all but finitely many Dirichlet characters \(\chi \) of p-power conductor.
