On special values of standard L-functions of Siegel cusp eigenforms of genus 3

2013/12/23 by Do, Anh Tuan, Vankov, Kirill
#11F46 #11F67 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1312.6623

Abstract

We explicitly compute the special values of the standard L-function L(s, F₁₂, St) at the critical points s∈\-8, -6, -4, -2, 0, 1, 3, 5, 7, 9\, where F₁₂ is the unique (up to a scalar) Siegel cusp form of degree 3 and weight 12, which was constructed by Miyawaki. These values are proportional to the product of the Petersson norms of symmetric square of Ramanujan's Δ and the cusp form of weight 20 for \rm SL₂(ℤ) by a rational number and some power of π. We use the Rankin-Selberg method and apply the Holomorphic projection to compute these values. To our knowledge this is the first example of a standard L-function of Siegel cusp form of degree 3, when the special values can be computed explicitly.

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