| L(s) = 1 | + 5.66·2-s − 95.8·4-s + 15.0·5-s + 499.·7-s − 1.26e3·8-s + 85.0·10-s − 5.25e3·11-s + 114.·13-s + 2.82e3·14-s + 5.08e3·16-s + 1.67e4·17-s + 5.40e3·19-s − 1.43e3·20-s − 2.97e4·22-s + 7.46e4·23-s − 7.78e4·25-s + 649.·26-s − 4.78e4·28-s − 5.87e4·29-s + 1.94e5·31-s + 1.91e5·32-s + 9.50e4·34-s + 7.49e3·35-s + 3.19e5·37-s + 3.06e4·38-s − 1.90e4·40-s − 1.72e5·41-s + ⋯ |
| L(s) = 1 | + 0.500·2-s − 0.749·4-s + 0.0537·5-s + 0.550·7-s − 0.876·8-s + 0.0269·10-s − 1.18·11-s + 0.0144·13-s + 0.275·14-s + 0.310·16-s + 0.827·17-s + 0.180·19-s − 0.0402·20-s − 0.595·22-s + 1.27·23-s − 0.997·25-s + 0.00724·26-s − 0.412·28-s − 0.447·29-s + 1.17·31-s + 1.03·32-s + 0.414·34-s + 0.0295·35-s + 1.03·37-s + 0.0905·38-s − 0.0470·40-s − 0.390·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 + 2.05e5T \) |
| good | 2 | \( 1 - 5.66T + 128T^{2} \) |
| 5 | \( 1 - 15.0T + 7.81e4T^{2} \) |
| 7 | \( 1 - 499.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 5.25e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 114.T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.67e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 5.40e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 7.46e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 5.87e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.94e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.19e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 1.72e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.74e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + 7.01e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 4.67e5T + 1.17e12T^{2} \) |
| 61 | \( 1 + 8.78e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.01e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 7.22e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 6.53e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.78e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.90e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + 2.70e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.47e7T + 8.07e13T^{2} \) |
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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
−9.346967474160349570235106158670, −8.208133902819168622658487084286, −7.71779937417456937265561901356, −6.27953833125494678866624160394, −5.26705545277994810886416453888, −4.78273265828931622840450913217, −3.57341325004856498776678633342, −2.62435733852096874489637962203, −1.15365121581910865240978523350, 0,
1.15365121581910865240978523350, 2.62435733852096874489637962203, 3.57341325004856498776678633342, 4.78273265828931622840450913217, 5.26705545277994810886416453888, 6.27953833125494678866624160394, 7.71779937417456937265561901356, 8.208133902819168622658487084286, 9.346967474160349570235106158670
