Taking the central value – 1.5 (between – 1 and – 2),
ζ(– 1.5) = – .025485...
This is well approximated by – e ^{– π(k)}^{ }* 2^{k }= – .025408... (where k = 1.5)
I then attempted to obtain approximations based on this formula replacing the power (or exponent) with respect to both e and 2 with the corresponding value of k.
This works well down to k = 1.
However it is much more problematic in the other direction.
Here rather than changing k successively upwards from 1.6 to 2, it works better to decrease in again reaching k = 1 for s = – 2.
However even here the approximation proves very inaccurate as we approach s = – 2, where ζ(– 2) = 0.
Once again I give a table showing values for s, ζ(s), approximation for ζ(s) and % relative accuracy.
s

ζ(s)

Approximation

% relative accuracy

– 1.1

– .067981

– .067657

99.5

– 1.2

– .054788

– .052964

96.7

– 1.3

– .043464

– .041461

95.4

– 1.4

– .033764

– .032457

96.1

– 1.5

– .025485

– .025408

99.7

– 1.6

– .018448

– .017315

93.9

– 1.7

– .012505

– .011800

94.3

– 1.8

– .007522

– .008041

93.5

– 1.9

– .003387

– .005480

61.8

– 2.0

0

– .003734

0

With the exception of the final two values, the approximations are quite accurate. It seems that the negative even values for s, where ζ(s) = 0, operate as attractors in their vicinity, upsetting previous trends.
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