Signs

The computation of the Riemann zeta function involves knowing
more than the identities on which it rests. In effect, the zeroes in
question are those that appear for S on the real line! As one person put it in
a forum: Computing the Riemann zeta function is an entire new branch
of mathematics.

Riemann was a student of Dirichlet's; we owe him the modern definition of
the function as a single-valued correspondance between domain and range.
Dirichlet also opened the study of L-functions, of which the Riemann zeta
is the simplest expression.

http://www.lmfdb.org/

Below, a look at sign changes:

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As for why the Geogebra numbers move the way they do...tomorrow!

I'll be starting here:

source: Sciencedirect