| L(s) = 1 | + 2.17·2-s + 2.73·4-s − 5-s + 0.443·7-s + 1.59·8-s − 2.17·10-s + 1.59·11-s + 0.964·14-s − 1.99·16-s − 3.76·17-s − 1.54·19-s − 2.73·20-s + 3.46·22-s − 0.705·23-s + 25-s + 1.21·28-s − 2.49·29-s − 6.18·31-s − 7.53·32-s − 8.19·34-s − 0.443·35-s − 10.3·37-s − 3.36·38-s − 1.59·40-s + 6.02·41-s + 4.72·43-s + 4.35·44-s + ⋯ |
| L(s) = 1 | + 1.53·2-s + 1.36·4-s − 0.447·5-s + 0.167·7-s + 0.563·8-s − 0.687·10-s + 0.480·11-s + 0.257·14-s − 0.499·16-s − 0.913·17-s − 0.354·19-s − 0.610·20-s + 0.738·22-s − 0.147·23-s + 0.200·25-s + 0.228·28-s − 0.464·29-s − 1.11·31-s − 1.33·32-s − 1.40·34-s − 0.0749·35-s − 1.70·37-s − 0.545·38-s − 0.251·40-s + 0.940·41-s + 0.720·43-s + 0.655·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.17T + 2T^{2} \) |
| 7 | \( 1 - 0.443T + 7T^{2} \) |
| 11 | \( 1 - 1.59T + 11T^{2} \) |
| 17 | \( 1 + 3.76T + 17T^{2} \) |
| 19 | \( 1 + 1.54T + 19T^{2} \) |
| 23 | \( 1 + 0.705T + 23T^{2} \) |
| 29 | \( 1 + 2.49T + 29T^{2} \) |
| 31 | \( 1 + 6.18T + 31T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 - 6.02T + 41T^{2} \) |
| 43 | \( 1 - 4.72T + 43T^{2} \) |
| 47 | \( 1 - 5.41T + 47T^{2} \) |
| 53 | \( 1 + 5.51T + 53T^{2} \) |
| 59 | \( 1 - 8.72T + 59T^{2} \) |
| 61 | \( 1 + 9.73T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 + 15.1T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 + 9.12T + 83T^{2} \) |
| 89 | \( 1 - 4.75T + 89T^{2} \) |
| 97 | \( 1 + 8.41T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.19191795374089267063826570370, −6.74366417494478911818233328607, −5.95455193576099243141917677583, −5.34982996857499241834664722974, −4.55097047358203881129619948739, −4.03564107565263184448862799046, −3.41899446381875465190826166948, −2.50703188017614302076516306832, −1.65848245701307417889341431677, 0,
1.65848245701307417889341431677, 2.50703188017614302076516306832, 3.41899446381875465190826166948, 4.03564107565263184448862799046, 4.55097047358203881129619948739, 5.34982996857499241834664722974, 5.95455193576099243141917677583, 6.74366417494478911818233328607, 7.19191795374089267063826570370
