| L(s) = 1 | − 1.27·2-s − 2.40·3-s − 0.372·4-s − 5-s + 3.07·6-s + 7-s + 3.02·8-s + 2.79·9-s + 1.27·10-s + 3.46·11-s + 0.897·12-s + 3.32·13-s − 1.27·14-s + 2.40·15-s − 3.11·16-s + 1.57·17-s − 3.56·18-s − 3.81·19-s + 0.372·20-s − 2.40·21-s − 4.41·22-s + 7.52·23-s − 7.28·24-s + 25-s − 4.24·26-s + 0.488·27-s − 0.372·28-s + ⋯ |
| L(s) = 1 | − 0.902·2-s − 1.39·3-s − 0.186·4-s − 0.447·5-s + 1.25·6-s + 0.377·7-s + 1.07·8-s + 0.932·9-s + 0.403·10-s + 1.04·11-s + 0.259·12-s + 0.923·13-s − 0.340·14-s + 0.621·15-s − 0.778·16-s + 0.382·17-s − 0.840·18-s − 0.875·19-s + 0.0833·20-s − 0.525·21-s − 0.941·22-s + 1.56·23-s − 1.48·24-s + 0.200·25-s − 0.833·26-s + 0.0940·27-s − 0.0704·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 + 1.27T + 2T^{2} \) |
| 3 | \( 1 + 2.40T + 3T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 - 3.32T + 13T^{2} \) |
| 17 | \( 1 - 1.57T + 17T^{2} \) |
| 19 | \( 1 + 3.81T + 19T^{2} \) |
| 23 | \( 1 - 7.52T + 23T^{2} \) |
| 29 | \( 1 + 6.34T + 29T^{2} \) |
| 31 | \( 1 - 5.51T + 31T^{2} \) |
| 37 | \( 1 + 0.599T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 + 4.41T + 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 - 0.322T + 59T^{2} \) |
| 61 | \( 1 + 2.75T + 61T^{2} \) |
| 67 | \( 1 - 7.46T + 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 + 3.40T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 + 8.83T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−7.47822072153303079004240326377, −6.81297574512939429514112843634, −6.20783587564599672603804532050, −5.45533968573704252538391782759, −4.60985727415345997632924067750, −4.22982934044473911349473586942, −3.19393939896746783853321520710, −1.55681381503508666068119986614, −1.03774788953884503186768837894, 0,
1.03774788953884503186768837894, 1.55681381503508666068119986614, 3.19393939896746783853321520710, 4.22982934044473911349473586942, 4.60985727415345997632924067750, 5.45533968573704252538391782759, 6.20783587564599672603804532050, 6.81297574512939429514112843634, 7.47822072153303079004240326377
