The spectrum of an element of the convolution algebra of a nonabelian group

Question

Let $G$ be a locally compact group and $L^1(G)$ its convolution algebra. If $G$ is Abelian, then the spectrum of an element $f \in L^1(G)$ is equal to the image of $\hat{f}$, the Fourier transform of $f$. Is there a similar nice description for the spectrum of $f$ when $G$ is not assumed to be Abelian? If it helps, we can suppose $G$ is compact.