L-Functions

Dirichlet series, zeros on the critical line, and the Riemann Hypothesis — visualized through domain coloring

30
2.0
First zero14.1347...
Second zero21.0220...
Third zero25.0109...
L(s,χ) = Σ χ(n)/n^s = Π_p (1 - χ(p)/p^s)^{-1}

Domain coloring maps the complex value to hue (argument) and brightness (modulus). Zeros appear as points where all colors meet. The Riemann Hypothesis asserts all non-trivial zeros of ζ(s) lie on Re(s) = 1/2.