E-008 — Li coefficients n<=2000 from zero table with tail bounds
08-17-2026 · 2000 rows · script
research/numerics/scripts/E-008-li-coefficients.py
Metadata
Li coefficients lambda_n (n<=2000) from E-002 zeros with explicit tail bound; validation vs lambda_1 literature value; leading-asymptotic ratio trend
Validation checks
| lambda_1_within_tail_bound | PASS |
| all_positive_in_range | PASS |
| usable_range_at_least_n100 | PASS |
Key values
| lambda_1 | 0.022961009776646203 |
| lambda_100 | 117.25678701049617 |
| lambda_1000 | 2191.3892531979554 |
| lambda_2000 | 4813.526935739563 |
Parameters
| n_max | 2000 |
| zeros_used | 10000 |
| H | 9877.782654005501 |
| precision | float64 |
Caveats
- truncated at H; tail bound explicit in CSV; usable range (tail<5%) is only n<=~116 with 10^4 zeros — n~10^5 requires the prime-side Bombieri-Lagarias exact formulas (noted as task 3.2 continuation)
- positivity on our data is trivial (zeros on-line) — not evidence about RH
Chart
Data preview (2000 rows total)
| n | lambda_n_trunc | tail_bound | lambda_over_half_n_log_n |
|---|---|---|---|
| 1 | 0.02296100978 | 0.00053881 | |
| 2 | 0.09180693847 | 0.00215524 | 0.13244941485069064 |
| 3 | 0.2064266279 | 0.00484929 | 0.12526507606054607 |
| 4 | 0.3666352925 | 0.00862096 | 0.13223572956501617 |
| 5 | 0.5721752347 | 0.0134702 | 0.1422049848148852 |
| 6 | 0.8227168415 | 0.0193972 | 0.15305567062901207 |
| 7 | 1.117859857 | 0.0264017 | 0.16413323648684855 |
| 8 | 1.457134929 | 0.0344838 | 0.17518344468412358 |
| 9 | 1.840005414 | 0.0436436 | 0.18609390058540218 |
| 10 | 2.265869445 | 0.053881 | 0.19681091930016117 |
| 11 | 2.734062237 | 0.065196 | 0.20730772962145133 |
| 12 | 3.243858638 | 0.0775886 | 0.21757079139798177 |
| 13 | 3.794475896 | 0.0910589 | 0.2275933911665634 |
| 14 | 4.385076639 | 0.105607 | 0.23737245599476084 |
| 15 | 5.014772056 | 0.121232 | 0.24690689776885238 |
| 16 | 5.682625259 | 0.137935 | 0.25619672749715683 |
| 17 | 6.387654809 | 0.155716 | 0.2652425743617933 |
| 18 | 7.128838405 | 0.174574 | 0.2740454247458607 |
| 19 | 7.905116691 | 0.19451 | 0.2826064837346345 |
| 20 | 8.715397202 | 0.215524 | 0.290927105832316 |
| 21 | 9.558558387 | 0.237615 | 0.2990087649745957 |
| 22 | 10.43345374 | 0.260784 | 0.3068530466490012 |
| 23 | 11.33891595 | 0.28503 | 0.31446165208036386 |
| 24 | 12.27376117 | 0.310355 | 0.3218364085460194 |
| 25 | 13.23679323 | 0.336756 | 0.328979282293977 |