  We compute the special values for the spinor L-function L(s,F12) in the
critical strip s={12,...,19}, 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 Petersson inner products of
Ramanujan's Delta by itself and the cusp form of weight 20 for SL(2,Z) by
itself by a rational number and some power of Pi. We also verify this result
numerically using Dokchitser's ComputeL PARI package. To our knowledge this is
the first example of a spinor L-function of Siegel cusp forms of degree 3, when
the special values can be computed explicitly.
