| L(s) = 1 | + 2.77·3-s + 1.39·5-s − 3.94·7-s + 4.67·9-s − 1.21·11-s − 2.11·13-s + 3.86·15-s + 17-s − 5.93·19-s − 10.9·21-s + 5.14·23-s − 3.04·25-s + 4.64·27-s − 9.48·29-s + 2.73·31-s − 3.37·33-s − 5.51·35-s + 9.34·37-s − 5.86·39-s + 5.18·41-s − 3.21·43-s + 6.52·45-s − 2.45·47-s + 8.57·49-s + 2.77·51-s − 4.85·53-s − 1.70·55-s + ⋯ |
| L(s) = 1 | + 1.59·3-s + 0.624·5-s − 1.49·7-s + 1.55·9-s − 0.367·11-s − 0.586·13-s + 0.998·15-s + 0.242·17-s − 1.36·19-s − 2.38·21-s + 1.07·23-s − 0.609·25-s + 0.893·27-s − 1.76·29-s + 0.491·31-s − 0.587·33-s − 0.931·35-s + 1.53·37-s − 0.938·39-s + 0.810·41-s − 0.490·43-s + 0.973·45-s − 0.357·47-s + 1.22·49-s + 0.387·51-s − 0.666·53-s − 0.229·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 - 2.77T + 3T^{2} \) |
| 5 | \( 1 - 1.39T + 5T^{2} \) |
| 7 | \( 1 + 3.94T + 7T^{2} \) |
| 11 | \( 1 + 1.21T + 11T^{2} \) |
| 13 | \( 1 + 2.11T + 13T^{2} \) |
| 19 | \( 1 + 5.93T + 19T^{2} \) |
| 23 | \( 1 - 5.14T + 23T^{2} \) |
| 29 | \( 1 + 9.48T + 29T^{2} \) |
| 31 | \( 1 - 2.73T + 31T^{2} \) |
| 37 | \( 1 - 9.34T + 37T^{2} \) |
| 41 | \( 1 - 5.18T + 41T^{2} \) |
| 43 | \( 1 + 3.21T + 43T^{2} \) |
| 47 | \( 1 + 2.45T + 47T^{2} \) |
| 53 | \( 1 + 4.85T + 53T^{2} \) |
| 61 | \( 1 + 4.09T + 61T^{2} \) |
| 67 | \( 1 + 7.73T + 67T^{2} \) |
| 71 | \( 1 - 8.39T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 - 9.67T + 79T^{2} \) |
| 83 | \( 1 - 1.15T + 83T^{2} \) |
| 89 | \( 1 + 5.99T + 89T^{2} \) |
| 97 | \( 1 + 17.6T + 97T^{2} \) |
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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
−7.61840126176599597214762470918, −6.87518586883002743234107142678, −6.23271092848705547824116367376, −5.49959411317100457589028666206, −4.37480022800767703422647083073, −3.73553727121149636622863675115, −2.85085367805349667452281063722, −2.55991454043041972569088235298, −1.61241798454744726494844724035, 0,
1.61241798454744726494844724035, 2.55991454043041972569088235298, 2.85085367805349667452281063722, 3.73553727121149636622863675115, 4.37480022800767703422647083073, 5.49959411317100457589028666206, 6.23271092848705547824116367376, 6.87518586883002743234107142678, 7.61840126176599597214762470918
