The question is in the title: The book


Godement-Jacquet "Zeta functions of simple algebras"


is from 1971. Has there ever been a textbook introduction to this material, or at least part of it? (but beyond just Tate's thesis).



Contemporary introduction to Godement-Jacquet "Zeta functions of simple algebras"
As you mention in your question, for $\mathrm{GL}\_1$, this is just Tate's thesis revisited.


The only exposition on this that I know of occurs in "[Automorphic Representations and $L$-Functions for the General Linear Group](https://doi.org/10.1017/CBO9780511973628)" by Dorian Goldfeld and Joseph Hundley. In particular, Chapter 11 of volume 1 covers the theory of the Godement-Jacquet zeta integral for $\mathrm{GL}\_2(\mathbb{A}\_{\mathbb{Q}})$ in some detail. Chapter 15 of volume 2 covers the same theory for $\mathrm{GL}\_n(\mathbb{A}\_{\mathbb{Q}})$ with $n$ arbitrary.


More generally, the two volumes are a detailed introduction to the theory of automorphic representations of $\mathrm{GL}\_n(\mathbb{A}\_{\mathbb{Q}})$ at a reasonable pace. There are other books that cover the $\mathrm{GL}\_2$ theory at a little higher level (for example, "[Automorphic Forms and Representations](https://doi.org/10.1017/CBO9780511609572)" by Daniel Bump or the original, "[Automorphic Forms on $\mathrm{GL}(2)$](http://sunsite.ubc.ca/DigitalMathArchive/Langlands/pdf/jl-ps.pdf)" by Hervé Jacquet and Robert Langlands) or at a quicker pace ("[Automorphic Forms on Adele Groups](https://www.jstor.org/stable/j.ctt1b7x82z)" by Stephen Gelbart). To my knowledge, however, Goldfeld-Hundley is the only textbook that covers the Godement-Jacquet zeta integral in detail.


I should also mention that Hervé Jacquet has also written two short surveys on the Godement-Jacquet theory: "[Principal $L$-Functions of the Linear Group](https://doi.org/10.1090/pspum/033.2)", which appears in the famous Corvallis proceedings, and "[Principal $L$-Functions for $\mathrm{GL}(n)$](https://doi.org/10.1090/pspum/061)".




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Edit: per the [review of Goldfeld-Hundley](https://doi.org/10.1090/S0273-0979-2013-01399-5) by Ramin Takloo-Bighash in the Bulletin of the American Mathematical Society, "The book under review is the first book since the appearance of [Godement-Jacquet] in 1972 that gives a thorough treatment of the principal $L$-function using matrix coefficients."




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Second edit: Google Scholar has informed me that there is a new book coming out soon, [*The Genesis of the Langlands Program*](https://www.cambridge.org/core/books/genesis-of-the-langlands-program/FDE4BFC05DF4074D04520FDF9CC67A4E), that includes a long chapter by Goldfeld and Jacquet called "Automorphic Representations and $L$-Functions for $\mathrm{GL}(n)$", which seems to be a detailed discussion of the theory of the Godement-Jacquet zeta integral. You can read some of this chapter on [Google Books](https://books.google.com/books?hl=en&lr=&id=ZW8yEAAAQBAJ&oi=fnd&pg=PA215&ots=NJ9-Y8i7WS&sig=2O7GqCckomWnEGw-8ir3Z1_3Nxk#v=onepage&q&f=false).


