The Langlands-Shahidi method says that the $L$-functions of automorphic representations can appear in the constant terms of Eisenstein series. Since those Eisenstein series have analytic continuation and functional equation, the $L$-function also inherits those nice properties. Most expositions of the Langlands-Shahidi method are phrased in the adelic language, which I am not too comfortable with. So I would like to understand it in a more classical language, without using adeles.

My question is: Suppose you have a weight $2$ eigenform $f \in S_2(\Gamma_0(N))$. Consider the $L$-function $L(f,s)$ attached to $f$. Then what is the Eisenstein series whose constant term contains the $L$-function $L(f,s)$? How can the nice analytic properties of $L(f,s)$ be read off from the analytic properties of the Eisenstein series? Is there a reference that explains this in classical language, without adeles?

The Langlands-Shahidi method says that the $L$-functions of automorphic representations appear in the constant terms of Eisenstein series. Since those Eisenstein series have analytic continuation and functional equation, the $L$-function also inherits those nice properties. 

Most expositions of the Langlands-Shahidi method are phrased in the adelic language, which I am not too comfortable with. So I would like to understand it in a more classical language, without using adeles.

My question is: Suppose you have a weight $2$ eigenform $f \in S_2(\Gamma_0(N))$. Consider the $L$-function $L(f,s)$ attached to $f$. Is there a reference that explains the Langlands-Shahidi method for $L(f,s)$ without referring to adeles? Namely, in classical language, what is the Eisenstein series whose constant term contains the $L$-function $L(f,s)$? How can the nice analytic properties of $L(f,s)$ be read off from the analytic properties of the Eisenstein series?