Andrew Booker introduced a framework to study L-functions through distributions in https://arxiv.org/abs/1308.3067v2. This allowed him and others to get new results about zeros of automorphic L-functions, like the existence of infinitely zeros of $GL(3)$ L-functions of odd order. He also conjectures in this talk the existence of an algebraic structure for the set of L-functions.
Have there been recent developments of this approach leading to new insights about L-functions? Moreover, as the Fourier transform is an automorphism of the Schwartz class, does it give rise to some Fourier transform based symmetry of L-functions?
