The distribution of a product of two normally distributed variates and with zero means and variances and is given by
(1)
 
(2)

where is a delta function and is a modified Bessel function of the second kind. This distribution is plotted above in red.
The analogous expression for a product of three normal variates can be given in terms of Meijer Gfunctions as
(3)

plotted above in blue.