0.1. The Notion of Automorphic L-functions. The notion of automorphic
L-functions, which was introduced by R. Langlands in 1960's, generalizes that of zeta
functions of global fields (number fields or function fields) and Hecke L-functions as-
sociated to classical modular forms. By conjectured Reciprocity Laws, L-functions
attached to Galois representations, algebraic varieties, (co)homology groups, mo-
tives, and automorphic forms should become special cases of automorphic L-functions.
Let us recall the notion of automorphic L-functions, which can be found in [Lan3],
[Borl],[GeSh], and [ArGe].
For the sake of simplicity, we assume that F is a number field and G is a connected
reductive algebraic group splitting over F. As usual, we let G& be the adele group of
G and LG the L-group associated to G. We can assume that the L-group LG is the
complex dual group of G since we only consider the groups splitting over F.
Given an irreducible automorphic cuspidal representation ir = SVTTV of G& and a
finite dimensional representation r of the dual group L G, it is known that for almost
every finite place v, TTV is unramified. When TTV is unramified, it determines via the
Satake isomorphism a semi-simple conjugacy class TV in the local group LGV and the
local Langlands L-factor is defined by
L(s,7rw,rv) = [det(I - r ^ ) * ? " 5 ) ] - 1
where qv is the cardinality of the residue class field of Fv and rv = r o rjv is the
corresponding representation of LGV, the L-group of G as a group over Fv, obtained
by composing r with the natural homomorphism rjv : LGV — LG.
Given the data (G, 7r, r), there exists a global L-function introduced by Langlands
as follows, which is as usual called the automorphic L-function attached to the data
L5(s,7r,r) = Y[L(s,7rv,rv)
where S is such a finite set of places of F that if a place v of F is not in 5, then
v is finite and irv is unramified. Langlands proved that the automorphic L-function
Ls(s,ir,r) converges absolutely for Re(s) large and made the following conjecture on
the analytic properties of the L-function.