Integral Representations of Riemann auxiliary function

We prove that the auxiliary function R(s) has the integral representation \[{ R}(s)=-{2^s πˢeπ i s/4}{Γ(s)}∫_0^∞ yˢ{1-e-π y^2+π ω y}{1-e2π ω y}\,dy/y, ω=eπ i/4, s>0,\] valid for σ>0. The function in the integrand 1-e-π y²+π ω y1-e2π ω y is entire. Therefore, no residue is added when we move the path of integration.