| L(s) = 1 | + 1.70·2-s + 3.59·3-s − 5.09·4-s + 7.48·5-s + 6.13·6-s − 22.3·8-s − 14.0·9-s + 12.7·10-s + 34.9·11-s − 18.3·12-s + 23.9·13-s + 26.9·15-s + 2.62·16-s + 84.3·17-s − 23.9·18-s − 57.3·19-s − 38.0·20-s + 59.6·22-s − 154.·23-s − 80.3·24-s − 69.0·25-s + 40.8·26-s − 147.·27-s − 109.·29-s + 45.8·30-s + 60.9·31-s + 183.·32-s + ⋯ |
| L(s) = 1 | + 0.603·2-s + 0.692·3-s − 0.636·4-s + 0.669·5-s + 0.417·6-s − 0.986·8-s − 0.520·9-s + 0.403·10-s + 0.958·11-s − 0.440·12-s + 0.510·13-s + 0.463·15-s + 0.0410·16-s + 1.20·17-s − 0.314·18-s − 0.692·19-s − 0.425·20-s + 0.578·22-s − 1.40·23-s − 0.683·24-s − 0.552·25-s + 0.307·26-s − 1.05·27-s − 0.701·29-s + 0.279·30-s + 0.353·31-s + 1.01·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 - 1.70T + 8T^{2} \) |
| 3 | \( 1 - 3.59T + 27T^{2} \) |
| 5 | \( 1 - 7.48T + 125T^{2} \) |
| 11 | \( 1 - 34.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 23.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 84.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 57.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 154.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 109.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 60.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 197.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 374.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 492.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 65.4T + 1.03e5T^{2} \) |
| 53 | \( 1 + 426.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 353.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 601.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 49.6T + 3.00e5T^{2} \) |
| 71 | \( 1 - 45.2T + 3.57e5T^{2} \) |
| 73 | \( 1 + 757.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 513.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 99.6T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.13e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 853.T + 9.12e5T^{2} \) |
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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
−8.431139376124589764379873971702, −7.62672602155060710731860026938, −6.27749000385351760159604714631, −5.92772407034899461065722656109, −5.09391199723167762598515131536, −3.89122130642130941140209408720, −3.58975687251815660770016940320, −2.45175355677186134232363647934, −1.43674300144431397114894134031, 0,
1.43674300144431397114894134031, 2.45175355677186134232363647934, 3.58975687251815660770016940320, 3.89122130642130941140209408720, 5.09391199723167762598515131536, 5.92772407034899461065722656109, 6.27749000385351760159604714631, 7.62672602155060710731860026938, 8.431139376124589764379873971702
