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I love reading John Baez, he makes me feel like I can almost understand this stuff.
You can head over to cocalc.com, fire up a sage worksheet, and stick this in:

    for n in range(1, 13):
        field = GF(2^n)
        count = 0
        for x in field:
          for y in field:
            if y^2+y == x^3+x:
                count += 1
        print n, 2^n, count
    
if you want to play around with some of these numbers.
If you're interested in a lengthier intro to the Riemann Hypothesis, I highly recommend the book "The Millenium Problems" by Keith Devlin.