On a theorem of Levinson

Levinson investigated the number of real zeros of the real or imaginary part of π^-σ/2-it/2Γ σ/2 + it/2 ζ(σ+it), where σ>0 and ζ(s) is the Riemann zeta function. By the functional equation, π^-s/2 Γ s/2 ζ(s)=π^-1-s/2 Γ1-s/2 ζ(1-s), we may assume σ 1/2. In this paper, we consider π^-s+λ/2Γ s+λ/2 ζ(s+λ) ±π^-s-λ/2Γ s-λ/2 ζ (s-λ) 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.