Probability density function for other norm? ($\ell_2$ norm is normal distribution)

Question

For a vector, we know that the standard normal distribution has pdf:$$f(x) = \frac{1}{(2\pi)^{n/2}}\exp(-\frac{||x-\mu||_2^2}2),$$

which depends on the $\ell_2$ norm $||x-\mu||_2^2$. Can we have other pdfs which have the similar form but depend on other norms like $\ell_1$ and $\ell_\infty$?