Continuous crop circles drawn by Riemann's zeta function

Let η(s)=∑n=1∞(−1)n+1n−s be the alternating zeta function. For a real number τ we define certain complex numbers bM,m(τ) and consider finite Dirichlet series υM(τ,s)=∑m=1MbM,m(τ)m−s and ηN(τ,s)=∑M=1NυM(τ,s). Computations demonstrate some remarkable properties of these finite Dirichlet series, but nothing was supported by a proof so far.