I'm giving a talk today on modular forms and their $L$-functions at the Simons Center for Geometry and Physics. The slides for this talk are available here.
I refer to many things that I have done before in this talk.
For references and proofs of the various aspects of Dirichlet series associated to half-integral weight modular forms, I wrote a series of notes stemming from this post.
For more information on computing and working with Maass forms, see notes from another talk I gave.
And finally, many of the objects discussed are in the LMFDB.
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