Survey paper "Riemann's Zeta Function and Beyond" by S. Gelbart and S.D. Miller, Bulletin of the AMS, 41, (2004), 59112. 

Group 
Representation of
Lgroup 
degree of
representation 
Integral
representation exists? 
LanglandsShahidi
method applies? 
Complete Lfunction
entire? 
Partial Lfunction
entire (taken
over unramified nonarchimedean cases)? 
Partial Lfunction
entire
(including archimedean places)? 
Ramified
calculations done
for integral representation/invariant pairing? 
GL(n) 
standard 
n 
Yes:
GodementJacquet, and later
JacquetPiatetskiShapiroShalika 
Yes 
Yes 
Yes 
Yes 
? 
GL(n) 
symmetric square 
n(n+1)/2 
Yes: Shimura (n=2,
holomorphic
forms), GelbartJacquet (general n=2), PiatetskiShapiroPatterson
(n=3), BumpGinzburg (general n); see also Kable 
Yes 
Unknown 
Yes 
Unknown 
Unknown 
GL(n) (n > 3) 
exterior square 
n(n1)/2 
Yes:
JacquetShalika and
BumpFriedberg Distributional pairing: MillerSchmid 
Yes 
Yes, if n=4 or odd;
unknown
otherwise 
Yes 
Yes, over Q 
For archimedean
spherical
principal series (Stade), and invariant pairings for
noncuspidal
local representations over Q 
GL(n)xGL(m) 
tensor product 
n*m 
JacquetShalikaPiatetskiShapiro 
Yes 
Yes (Moeglin Waldspurger) 
Yes 
Yes 
For integral
representations,
thought to be impossible at archimedean place except when m=n1,n, or
n+1; for invariant pairings, Yes for noncuspidal local representations
over Q 
GL(2)  symmetric Cube  4  
GL(3)  symmetric fourth power  5  
GL(3)  adjoint  8  
SO(7)  2nd fundamental representation of Sp(6)  14  
GSO(10)  spin  
GSO(12)  spin  
SO(n) x GL(k)  
GSp(4)  spin  4  
GSp(5)  standard  5  
GSp(6)  spin  
GSp(6) x GL(2)  spin x standard  
GSp(8)  spin  
GSp(10)  spin  
Sp(2n) x GL(k)  
E_{6}  standard  
E_{7}  standard  
E_{8}  
F_{4}  26  
G_{2} 