Some own properties of certain approximations of the alternating zeta function by finite Dirichlet series

We consider two particular methods of approximating the alternating zeta function (known also as Dirichlet eta function) by finite Dirichlet series. Numerical calculations indicate that such approximations have some remarkable properties connected with prime numbers and with the non-trivial zeros of the zeta function. However, these observations have not been supported so far by proofs. These properties are considered to be «own properties» of the approximations in the following sense: the infinite alternating Dirichlet series for the eta function does not possess counterparts of these properties.