| L(s) = 1 | + 1.11·3-s + 1.08·5-s − 0.863·7-s − 1.75·9-s + 0.657·11-s + 6.20·13-s + 1.21·15-s − 5.69·17-s − 6.45·19-s − 0.963·21-s − 3.81·25-s − 5.30·27-s + 0.101·29-s + 1.54·31-s + 0.733·33-s − 0.939·35-s + 7.33·37-s + 6.91·39-s + 6.09·41-s − 5.37·43-s − 1.91·45-s + 7.84·47-s − 6.25·49-s − 6.35·51-s + 4.75·53-s + 0.715·55-s − 7.19·57-s + ⋯ |
| L(s) = 1 | + 0.644·3-s + 0.486·5-s − 0.326·7-s − 0.585·9-s + 0.198·11-s + 1.71·13-s + 0.313·15-s − 1.38·17-s − 1.48·19-s − 0.210·21-s − 0.763·25-s − 1.02·27-s + 0.0189·29-s + 0.277·31-s + 0.127·33-s − 0.158·35-s + 1.20·37-s + 1.10·39-s + 0.951·41-s − 0.819·43-s − 0.284·45-s + 1.14·47-s − 0.893·49-s − 0.889·51-s + 0.653·53-s + 0.0964·55-s − 0.953·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 1.11T + 3T^{2} \) |
| 5 | \( 1 - 1.08T + 5T^{2} \) |
| 7 | \( 1 + 0.863T + 7T^{2} \) |
| 11 | \( 1 - 0.657T + 11T^{2} \) |
| 13 | \( 1 - 6.20T + 13T^{2} \) |
| 17 | \( 1 + 5.69T + 17T^{2} \) |
| 19 | \( 1 + 6.45T + 19T^{2} \) |
| 29 | \( 1 - 0.101T + 29T^{2} \) |
| 31 | \( 1 - 1.54T + 31T^{2} \) |
| 37 | \( 1 - 7.33T + 37T^{2} \) |
| 41 | \( 1 - 6.09T + 41T^{2} \) |
| 43 | \( 1 + 5.37T + 43T^{2} \) |
| 47 | \( 1 - 7.84T + 47T^{2} \) |
| 53 | \( 1 - 4.75T + 53T^{2} \) |
| 59 | \( 1 + 5.68T + 59T^{2} \) |
| 61 | \( 1 + 0.224T + 61T^{2} \) |
| 67 | \( 1 - 2.37T + 67T^{2} \) |
| 71 | \( 1 + 6.42T + 71T^{2} \) |
| 73 | \( 1 + 4.17T + 73T^{2} \) |
| 79 | \( 1 + 16.9T + 79T^{2} \) |
| 83 | \( 1 + 16.9T + 83T^{2} \) |
| 89 | \( 1 + 5.06T + 89T^{2} \) |
| 97 | \( 1 - 9.83T + 97T^{2} \) |
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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
−7.54139039941900996208302569150, −6.50734576368036627007593094491, −6.19049256714389319869448678792, −5.60698659507060141961627166710, −4.30199573678931749282667353619, −3.98692814122018533375152977644, −2.94829792545001854703680229421, −2.31149205417520633740516719395, −1.44477109885482923849980206994, 0,
1.44477109885482923849980206994, 2.31149205417520633740516719395, 2.94829792545001854703680229421, 3.98692814122018533375152977644, 4.30199573678931749282667353619, 5.60698659507060141961627166710, 6.19049256714389319869448678792, 6.50734576368036627007593094491, 7.54139039941900996208302569150
