312 FREYDOON SHAHID!, LANGLANDS-SHAHID! METHODUsing the standard formulawhere(2.30) then equals(2.31) f(l/2)f(s/2)/r((s + 1)/2).
Using f(l/2) = fa, (2.31) equals
71'-s/^2 r(s/2)/rr-(s+I/^2 lr(s + 1/2)
which is again L(s )/ L(s+ 1), where L(s) is the archimedean Hecke-Tate L-function
attached to the character lxls, the JR-component of our cusp form on T.
The main result of Langlands in [Lal] can be stated as follows. Let S be a finite
set of places with the property that if v r/. S, 71' v is an unramified representation.
Every f E V(s, 7r) is of the form
f E 0vEsV(s,7rv)@@v(tS{f~},where f~ is Kv-spherical for some S. To be precise, the decomposition 71' = 0v11'v
depends on a choice of M(Ov)-invariant vectors {xv} for all the unramified places
which one fixes once for all (cf. [SJ). The functions f~ must then satisfy f~ ( kv) = Xv
for all kv E Kv and all v r/. S. Assume further that f = 0vfv, with fv = f~ for all
v r/. S. For each i, let
(2.32) Ls(s,71',ri) =II L(s,7rv,ri )·
v(tS
Then by Lemma 2.1(2.33)
A(s, 7r)f(e) = (}] Ls(is, 71', i)/ Ls(l +is, 71', ri))
0v(tSf~(ev)@ 0vEsA(s, 11'v, Wo)fv(ev)·It now follows from the properties of the constant term A(s, 7r) that
Theorem 2.2 (Langlands [Lal]). The product quotient
m
II Ls(is, 71', ri)/ Ls(l +is, 71', ri)
i=l
is meromorphic on all of <C.Clearly one needs an induction to get this to lead to meromorphic continuation
of each L-function in the product. We will soon discuss this induction.