The same ball in two explorers: put ½ at the centre and the critical line becomes the strut through 0 and ∞, the functional equation a sign flip through the centre. Swap ζ for a function with the same symmetry and no Euler product (Davenport–Heilbronn) and the zeros step off the strut: symmetry alone does not pin them, the arithmetic is the extra incidence.
The base case. ζ painted on the Riemann sphere, 0 at the bottom, ∞ at the top, and 0, 1, ∞, −1 a quarter turn apart — 1 is half way to ∞. Three sliders turn it about its three struts; the tip about ±i has four stops whose shadows are the charts s/(1 − s), 2s − 1 (zeta’s orientation), 1 − 1/s (Li) and 1/(1 − 2s). The ball beside its shadow, with the six frame points and the zeros as beads.
+ open explorer → +ζ, ξ and Davenport–Heilbronn painted in the s-plane, in w = 1/(s − ½), and on the Riemann sphere with ½ in front; Hardy's Z(t) along the strut, with the off-line zero at t ≈ 85.7 marked.