{
  "id": "1312.6623",
  "version": "v1",
  "published": "2013-12-23T18:03:50.000Z",
  "updated": "2013-12-23T18:03:50.000Z",
  "title": "On special values of standard L-functions of Siegel cusp eigenforms of genus 3",
  "authors": [
    "Anh Tuan Do",
    "Kirill Vankov"
  ],
  "categories": [
    "math.NT"
  ],
  "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 ${\\rm SL}_2(\\mathbb{Z})$ 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-12-23T18:03:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F67",
      "11F46"
    ],
    "keywords": [
      "special values",
      "siegel cusp eigenforms",
      "standard l-functions",
      "siegel cusp form",
      "first example"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1312.6623D"
    }
  }
}