| L(s) = 1 | − 2.57·2-s + 2.82·3-s + 4.61·4-s + 1.34·5-s − 7.27·6-s − 3.61·7-s − 6.73·8-s + 4.98·9-s − 3.45·10-s + 11-s + 13.0·12-s − 0.296·13-s + 9.30·14-s + 3.79·15-s + 8.10·16-s − 4.19·17-s − 12.8·18-s + 0.592·19-s + 6.19·20-s − 10.2·21-s − 2.57·22-s + 1.33·23-s − 19.0·24-s − 3.19·25-s + 0.763·26-s + 5.61·27-s − 16.6·28-s + ⋯ |
| L(s) = 1 | − 1.81·2-s + 1.63·3-s + 2.30·4-s + 0.600·5-s − 2.96·6-s − 1.36·7-s − 2.38·8-s + 1.66·9-s − 1.09·10-s + 0.301·11-s + 3.76·12-s − 0.0822·13-s + 2.48·14-s + 0.979·15-s + 2.02·16-s − 1.01·17-s − 3.02·18-s + 0.135·19-s + 1.38·20-s − 2.22·21-s − 0.548·22-s + 0.278·23-s − 3.88·24-s − 0.639·25-s + 0.149·26-s + 1.08·27-s − 3.15·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - T \) |
| 547 | \( 1 - T \) |
| good | 2 | \( 1 + 2.57T + 2T^{2} \) |
| 3 | \( 1 - 2.82T + 3T^{2} \) |
| 5 | \( 1 - 1.34T + 5T^{2} \) |
| 7 | \( 1 + 3.61T + 7T^{2} \) |
| 13 | \( 1 + 0.296T + 13T^{2} \) |
| 17 | \( 1 + 4.19T + 17T^{2} \) |
| 19 | \( 1 - 0.592T + 19T^{2} \) |
| 23 | \( 1 - 1.33T + 23T^{2} \) |
| 29 | \( 1 - 5.99T + 29T^{2} \) |
| 31 | \( 1 - 7.96T + 31T^{2} \) |
| 37 | \( 1 + 7.24T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 + 3.89T + 43T^{2} \) |
| 47 | \( 1 + 5.62T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 + 6.98T + 59T^{2} \) |
| 61 | \( 1 + 1.94T + 61T^{2} \) |
| 67 | \( 1 + 8.97T + 67T^{2} \) |
| 71 | \( 1 - 2.79T + 71T^{2} \) |
| 73 | \( 1 - 1.22T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + 16.2T + 83T^{2} \) |
| 89 | \( 1 - 4.80T + 89T^{2} \) |
| 97 | \( 1 - 3.11T + 97T^{2} \) |
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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
−8.069599482527780527763269453241, −7.23263226716527758176633934350, −6.55200414050510217353500457077, −6.29297896939830703549298643437, −4.70489391628761291193222927743, −3.40124986322957835675636998051, −2.96383985974981081914239405209, −2.15613377016942028678780765209, −1.44848717059031854555476605143, 0,
1.44848717059031854555476605143, 2.15613377016942028678780765209, 2.96383985974981081914239405209, 3.40124986322957835675636998051, 4.70489391628761291193222927743, 6.29297896939830703549298643437, 6.55200414050510217353500457077, 7.23263226716527758176633934350, 8.069599482527780527763269453241
