| L(s) = 1 | − 2.08·2-s + 2.53·3-s + 2.36·4-s + 0.545·5-s − 5.30·6-s + 0.502·7-s − 0.761·8-s + 3.44·9-s − 1.13·10-s − 0.886·11-s + 6.00·12-s − 1.49·13-s − 1.04·14-s + 1.38·15-s − 3.13·16-s − 7.27·17-s − 7.19·18-s + 19-s + 1.28·20-s + 1.27·21-s + 1.85·22-s − 3.29·23-s − 1.93·24-s − 4.70·25-s + 3.12·26-s + 1.13·27-s + 1.18·28-s + ⋯ |
| L(s) = 1 | − 1.47·2-s + 1.46·3-s + 1.18·4-s + 0.243·5-s − 2.16·6-s + 0.189·7-s − 0.269·8-s + 1.14·9-s − 0.360·10-s − 0.267·11-s + 1.73·12-s − 0.415·13-s − 0.280·14-s + 0.357·15-s − 0.784·16-s − 1.76·17-s − 1.69·18-s + 0.229·19-s + 0.288·20-s + 0.278·21-s + 0.394·22-s − 0.686·23-s − 0.394·24-s − 0.940·25-s + 0.613·26-s + 0.217·27-s + 0.224·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
| good | 2 | \( 1 + 2.08T + 2T^{2} \) |
| 3 | \( 1 - 2.53T + 3T^{2} \) |
| 5 | \( 1 - 0.545T + 5T^{2} \) |
| 7 | \( 1 - 0.502T + 7T^{2} \) |
| 11 | \( 1 + 0.886T + 11T^{2} \) |
| 13 | \( 1 + 1.49T + 13T^{2} \) |
| 17 | \( 1 + 7.27T + 17T^{2} \) |
| 23 | \( 1 + 3.29T + 23T^{2} \) |
| 29 | \( 1 - 1.34T + 29T^{2} \) |
| 31 | \( 1 - 10.9T + 31T^{2} \) |
| 37 | \( 1 - 7.35T + 37T^{2} \) |
| 41 | \( 1 + 12.1T + 41T^{2} \) |
| 43 | \( 1 + 6.10T + 43T^{2} \) |
| 47 | \( 1 + 7.55T + 47T^{2} \) |
| 53 | \( 1 - 2.77T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 + 4.04T + 67T^{2} \) |
| 71 | \( 1 + 4.91T + 71T^{2} \) |
| 73 | \( 1 - 1.17T + 73T^{2} \) |
| 79 | \( 1 - 6.89T + 79T^{2} \) |
| 83 | \( 1 - 4.36T + 83T^{2} \) |
| 89 | \( 1 + 6.11T + 89T^{2} \) |
| 97 | \( 1 - 2.25T + 97T^{2} \) |
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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
−8.108404595339182102725948013062, −7.931460183918136577515241887007, −6.89988161331050664221162577662, −6.35027773025205300532238456484, −4.88385379591002557260784435803, −4.16313365641775827663405772646, −2.94710910923019270938648905329, −2.24809798536595127828048902159, −1.58222194099418400074826684535, 0,
1.58222194099418400074826684535, 2.24809798536595127828048902159, 2.94710910923019270938648905329, 4.16313365641775827663405772646, 4.88385379591002557260784435803, 6.35027773025205300532238456484, 6.89988161331050664221162577662, 7.931460183918136577515241887007, 8.108404595339182102725948013062
