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Algorithms to approximate numerically a root of a nonlinear equation or system: for instance, Newton's method, secant method, bisection, etc.
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Zeros of partial sums of the Ramanujan's zeta function
Here is how you can find reasonable answers using Mathematica (or WolframAlpha).
To see the first two zeros:
ContourPlot[ Log[Abs[1 - 24/2^(sig + I t) + 252/3^(sig + I t)]], {sig, 2, 7}, {t, 0, …