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CALCULATION OF DIRICHLET L-FUNCTIONS 491 (4) x(B3) = exp (27ri3j/hi), O < ?1 < hi; j = 1, **, R. Clearly, the R-tuple of nonnegative integers ,Bj determines and is determined by the character x. Using this representation, it is now easy to see that the characters form a group isomorphic to M(k) under the mapping x B10 – e -BROR, (where
Evaluation of Dirichlet L-functions 123 The L-functions, L(s, X), may be continued analytically over Ms < 1 and for primitive characters-they satisfy the functional equation L(8, X) = 2sk-sents-r(i -8) sin -1rsL(1 -s,-) (1.3) if 1) +1 and L(8, x) = 2sk-sers-lr(i -8 ) cos -7r8L(l -s, X) (1.4) if x(-1) =-1.
On Small Zeros of Dirichiet L–Functions By Peter J. Weinberger To D. H. Lehmer for his 70th birthday Abstract. A method is given for calculating the value of Dirichlet L–functions near the real axis in the critical strip. As an application, some zeros for zeta functions …
include as a special case (a = b = 1) the Riemann function c(s), characterized by Gram as "une des plus remarquables acquisitions de l‘analyse moderne"; while simple linear combinations of Z(a, b, s) for b = 1, 2, , a-1, when a is a fixed integer, give the Dirichlet L–functions. All of these special functions,
Approximate functional equation for Dirichlet L-functions 225 whzere NX = [V(t/2irk) + 2], R < At, c = X(a) cos lba where ~~~ ~~~~~k-1i7T a I1 and SI? = n02i+ n ? Epi! (2k) VJP)(20k(), with _~Sk kC with b-1 in(phx(n)-Phe?7r(2- 2 An k ) with Vr(A) = sin 17T(2A +k) n=2 For X(-1) =-1, we have exp( Qi) for exp(7 ri) in the third term of (11),