Approximation of the zeta function via finite Euler products

Consider finite Euler product
$$ \zeta_{m}(s)\,=\,\prod_{k=1}^{m}(1-p_k^{-s})^{-1}, $$
where $p_1,\,\dots,\,p_m$ are the initial primes, and finite xi function
$$ g(s)\,=\,\pi^{-\frac{s}{2}}(s-1)\Gamma\bigl({s}/{2}+1\bigr) $$
is the factor from the functional equation. Modified symmetrized finite xi function
$$ \xi^{ :=}_{{m}}(s) \! =\!s^m(1\!-\!s)^m\big(\xi_{m}(s)\!+\!\xi_{m}(1-s)\big) $$
trivially satisfies the functional equation $\xi^{ :=}_{{m}}(s)=\xi^{ :=}_{{m}}(1-s)$. All poles $q_1,\,q_2,\dots$ of this function are simple; let $r_1,\,r_2,\dots$ be corresponding residues, so the difference
$$ \xi^{ :\text{reg}=}_{{m}}(s)\,=\,\xi^{ :=}_{{m}}(s)-\sum_{k=1}^\infty r_k/(s-q_k). $$
Regularized finite Euler product
$$ \zeta^{\approx}_{{m}}(s)\,=\,\xi^{ :\text{reg}=}_{{m}}(s)/\big( s^m(1-s)^m g(s)\big) $$
gives surprisingly good approximations to the values and zeroes of the zeta function.
Example 1. The least (in absolute value) non-real zero of function $\zeta^{\approx}_{{1}}(s)$ (which is defined via single Euler factor $(1-2^{-s})^{-1}$) differs from the least non-trivial zero of the zeta function less than by $10^{-6}$.
Example 2. The three first Euler factors produce more than 30 correct decimal digits of $\zeta(1/2+100i)$.
For more examples visit
http://logic.pdmi.ras.ru/~yumat/personaljournal/
eulereverywhere.