The Riemann Hypothesis (1859, still unproven) asserts that all non-trivial zeros of ζ(s) lie on the critical line Re(s)=½. Here you can visualize ζ(s) on the complex plane and see the famous zeros where ζ vanishes.
Critical Line
ζ Magnitude
Argument Plot
Known zeros γₙ (Im part, ½+iγ):
γ₁ ≈ 14.1347
γ₂ ≈ 21.0220
γ₃ ≈ 25.0109
γ₄ ≈ 30.4249
γ₅ ≈ 32.9351
γ₆ ≈ 37.5862
γ₇ ≈ 40.9187
γ₈ ≈ 43.3271
γ₉ ≈ 48.0052
γ₁₀ ≈ 49.7738
γ₁₁ ≈ 52.9703
γ₁₂ ≈ 56.4462
γ₁₃ ≈ 59.3470
γ₁₄ ≈ 60.8318
γ₁₅ ≈ 65.1125
γ₁₆ ≈ 67.0798
γ₁₇ ≈ 69.5465
γ₁₈ ≈ 72.0672
γ₁₉ ≈ 75.7047
γ₂₀ ≈ 77.1448
Euler-Maclaurin approximation: We compute ζ(s) ≈ Σ n⁻ˢ using the first N terms (valid for Re(s)>1) with analytic continuation via the functional equation ζ(s)=2ˢπˢ⁻¹sin(πs/2)Γ(1−s)ζ(1−s). The critical line plot shows Re(ζ(½+it)) and Im(ζ(½+it)) — the curve spirals around the origin and crosses zero at each non-trivial zero.