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

Abstract

We explicitly compute the special values of the standard $L$-function $L(s,F_{12},\mathrm{St})$ at the critical points $s\in \{-8,-6,-4,-2,0,1,3,5,7,9\}$, where $F_{12}$ 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 $\Delta$ and the cusp form of weight $20$ for ${\mathrm{SL}}_2(\mathbb{Z})$ by a rational number and some power of $\pi$. 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.