On a theorem of Levinson

Levinson investigated the number of real zeros of the real or imaginary part of where σ>0 and ζ(s) is the Riemann zeta function. By the functional equation, we may assume . In this paper, we consider for any complex number s and any λ>0, as general forms of the real or imaginary part of the above function, and then we further study the zeros of the functions.