... particular, L(s, σ, r)isholomorphic for s>−1 except possibly for s =0and L(s, σ)isholomorphic for s>0. Therefore I(σ, s)isirreducible for 0 <s<1 except possibly for s =12and moreover, if ... the“standard” L-function ([GJ72]) L(s, π)isreal for s ∈and positive for s>1.Assuming GRH we have L(s, π) > 0 for 12<s≤ 1, except for the case wheren =1and π is the trivial character. ... the formXi(ωiνβi× ωiν−βi) with ωisquare-integrable and 0 <βi<12. Clearly, πn.t.appears as a factor of πnon-S-pairs.Wewill deform the nontempered parametersof π .For0 ≤...