On a theorem of Levinson

Levinson investigated the number of real zeros of the real or imaginary part of pi(-sigma/2-it/2)Gamma((sigma)/(2) + (it)/(2))zeta(sigma + it), where sigma>0 and zeta(s) is the Riemann zeta function. By the functional equation, pi(-slambda/2)Gamma((s+lambda)/(2))zeta(s+lambda)+/-pi(-s-lambda/2)Gamma((s-lambda)/(2))zeta(s-lambda) we may assume sigma > (1)/(2). In this paper, we consider pi(-s+lambda/2)Gamma((s+lambda)/(2))zeta(s+lambda)+/-pi(-s-lambda)/(2)Gamma((s-lambda)/(2))zeta(s-lambda) for any complex number s and any lambda > 0, as general forms of the real or imaginary part of the above function, and then we further study the zeros of the functions. (C) 2003 Elsevier Inc. All rights reserved.