| L(s) = 1 | − 2-s − 2.25·3-s + 4-s + 2.47·5-s + 2.25·6-s − 7-s − 8-s + 2.07·9-s − 2.47·10-s + 0.831·11-s − 2.25·12-s + 2.09·13-s + 14-s − 5.58·15-s + 16-s + 5.90·17-s − 2.07·18-s + 0.366·19-s + 2.47·20-s + 2.25·21-s − 0.831·22-s − 7.64·23-s + 2.25·24-s + 1.14·25-s − 2.09·26-s + 2.08·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.30·3-s + 0.5·4-s + 1.10·5-s + 0.919·6-s − 0.377·7-s − 0.353·8-s + 0.690·9-s − 0.784·10-s + 0.250·11-s − 0.650·12-s + 0.580·13-s + 0.267·14-s − 1.44·15-s + 0.250·16-s + 1.43·17-s − 0.488·18-s + 0.0841·19-s + 0.554·20-s + 0.491·21-s − 0.177·22-s − 1.59·23-s + 0.459·24-s + 0.229·25-s − 0.410·26-s + 0.402·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 + 2.25T + 3T^{2} \) |
| 5 | \( 1 - 2.47T + 5T^{2} \) |
| 11 | \( 1 - 0.831T + 11T^{2} \) |
| 13 | \( 1 - 2.09T + 13T^{2} \) |
| 17 | \( 1 - 5.90T + 17T^{2} \) |
| 19 | \( 1 - 0.366T + 19T^{2} \) |
| 23 | \( 1 + 7.64T + 23T^{2} \) |
| 29 | \( 1 + 3.72T + 29T^{2} \) |
| 31 | \( 1 + 0.559T + 31T^{2} \) |
| 37 | \( 1 + 4.02T + 37T^{2} \) |
| 41 | \( 1 + 5.95T + 41T^{2} \) |
| 43 | \( 1 - 2.73T + 43T^{2} \) |
| 47 | \( 1 + 0.309T + 47T^{2} \) |
| 53 | \( 1 - 0.908T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 + 5.62T + 67T^{2} \) |
| 71 | \( 1 + 2.39T + 71T^{2} \) |
| 73 | \( 1 + 2.80T + 73T^{2} \) |
| 79 | \( 1 + 6.06T + 79T^{2} \) |
| 83 | \( 1 + 3.04T + 83T^{2} \) |
| 89 | \( 1 - 9.61T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−7.68853408896122001612707439547, −6.82700262739663233419849434645, −6.20121043756274562283072161238, −5.71763791168558865392441711504, −5.31896675168525739094277954446, −4.06264368198712721430497086832, −3.12561764222363655396322948944, −1.92913316324780539611140259074, −1.19433817706390802986919829159, 0,
1.19433817706390802986919829159, 1.92913316324780539611140259074, 3.12561764222363655396322948944, 4.06264368198712721430497086832, 5.31896675168525739094277954446, 5.71763791168558865392441711504, 6.20121043756274562283072161238, 6.82700262739663233419849434645, 7.68853408896122001612707439547
