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We compute the special values for the spinor L-function L(s,F12) in thecritical strip s={12,...,19}, where F12 is the unique (up to a scalar) Siegelcusp form of degree 3 and weight 12, which was constructed by Miyawaki. Thesevalues are proportional to the product of Petersson inner products ofRamanujan's Delta by itself and the cusp form of weight 20 for SL(2,Z) byitself by a rational number and some power of Pi. We also verify this resultnumerically using Dokchitser's ComputeL PARI package. To our knowledge this isthe first example of a spinor L-function of Siegel cusp forms of degree 3, whenthe special values can be computed explicitly.

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