product of N normal distributions

#1
We are given N normal functions, with means \mu_i, and variances \sigma_i^2, i in 1...N.

I need to find a formula for the product of the N functions.

The case N=2 is easy to find online:

https://ccrma.stanford.edu/~jos/sasp/Product_Two_Gaussian_PDFs.html

but I have found nothing regarding the general case.

Because N=2 in normal, we know that the general result is normal, with a scaling factor. I need formulas for the scaling factor, the mean, and the variance.

Help is much appreciated!