On special values of standard \(L\)-functions of Siegel cusp eigenforms of genus 3. (English. French summary) Zbl 1380.11058
Summary: 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 I. Miyawaki [Mem. Fac. Sci., Kyushu Univ., Ser. A 46, No. 2, 307–339 (1992; Zbl 0780.11022)]. 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.
MSC:
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
| 11F46 | Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms |
Citations:
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