Formula / limit for 1/2+2*Pi/Log(2)

Clear[s, a, w, r];
f[s_, w_] = -(Integrate[1/(s - 1)*Exp[-2*Pi*I*s*w], s] -
    Integrate[1/(s - 1)*1/2^(s - 1)*Exp[-2*Pi*I*s*w], s])
w = 10^20;
Limit[f[s, w], s -> 1]
N[%, 30]^-1
im = Im[%]/w
re = Re[%%]