| L(s) = 1 | + 1.48·2-s − 7.08·3-s − 5.78·4-s + 7.50·5-s − 10.5·6-s + 9.60·7-s − 20.5·8-s + 23.1·9-s + 11.1·10-s − 27.6·11-s + 40.9·12-s + 10.8·13-s + 14.2·14-s − 53.1·15-s + 15.7·16-s + 17·17-s + 34.4·18-s + 111.·19-s − 43.4·20-s − 68.0·21-s − 41.1·22-s − 38.6·23-s + 145.·24-s − 68.6·25-s + 16.2·26-s + 27.1·27-s − 55.5·28-s + ⋯ |
| L(s) = 1 | + 0.525·2-s − 1.36·3-s − 0.723·4-s + 0.671·5-s − 0.716·6-s + 0.518·7-s − 0.906·8-s + 0.858·9-s + 0.353·10-s − 0.759·11-s + 0.986·12-s + 0.232·13-s + 0.272·14-s − 0.915·15-s + 0.246·16-s + 0.242·17-s + 0.451·18-s + 1.35·19-s − 0.485·20-s − 0.706·21-s − 0.399·22-s − 0.350·23-s + 1.23·24-s − 0.548·25-s + 0.122·26-s + 0.193·27-s − 0.375·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 - 17T \) |
| 43 | \( 1 + 43T \) |
| good | 2 | \( 1 - 1.48T + 8T^{2} \) |
| 3 | \( 1 + 7.08T + 27T^{2} \) |
| 5 | \( 1 - 7.50T + 125T^{2} \) |
| 7 | \( 1 - 9.60T + 343T^{2} \) |
| 11 | \( 1 + 27.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 10.8T + 2.19e3T^{2} \) |
| 19 | \( 1 - 111.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 38.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 41.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 218.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 78.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 208.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 388.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 742.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 132.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 451.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 366.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 669.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 365.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 457.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 767.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 28.9T + 7.04e5T^{2} \) |
| 97 | \( 1 + 890.T + 9.12e5T^{2} \) |
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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
−9.951118056443286155469932667673, −8.737909663111749102461502458991, −7.78885320045961366173961685996, −6.51694917157627987759238807155, −5.62895569452471214277256488496, −5.25293960473942722702163134328, −4.39905174570944547225729983036, −3.00102936367876986616752405414, −1.27943397047539974020235419774, 0,
1.27943397047539974020235419774, 3.00102936367876986616752405414, 4.39905174570944547225729983036, 5.25293960473942722702163134328, 5.62895569452471214277256488496, 6.51694917157627987759238807155, 7.78885320045961366173961685996, 8.737909663111749102461502458991, 9.951118056443286155469932667673
