| L(s) = 1 | − 2.67·2-s + 5.15·4-s − 0.519·5-s + 1.02·7-s − 8.42·8-s + 1.38·10-s − 5.14·11-s − 2.29·13-s − 2.73·14-s + 12.2·16-s + 1.92·17-s + 3.15·19-s − 2.67·20-s + 13.7·22-s + 6.68·23-s − 4.73·25-s + 6.13·26-s + 5.25·28-s + 7.68·29-s − 1.20·31-s − 15.8·32-s − 5.13·34-s − 0.530·35-s − 0.791·37-s − 8.42·38-s + 4.37·40-s − 2.49·41-s + ⋯ |
| L(s) = 1 | − 1.89·2-s + 2.57·4-s − 0.232·5-s + 0.385·7-s − 2.97·8-s + 0.439·10-s − 1.55·11-s − 0.636·13-s − 0.729·14-s + 3.05·16-s + 0.466·17-s + 0.722·19-s − 0.597·20-s + 2.93·22-s + 1.39·23-s − 0.946·25-s + 1.20·26-s + 0.993·28-s + 1.42·29-s − 0.216·31-s − 2.80·32-s − 0.881·34-s − 0.0895·35-s − 0.130·37-s − 1.36·38-s + 0.691·40-s − 0.389·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 239 | \( 1 + T \) |
| good | 2 | \( 1 + 2.67T + 2T^{2} \) |
| 5 | \( 1 + 0.519T + 5T^{2} \) |
| 7 | \( 1 - 1.02T + 7T^{2} \) |
| 11 | \( 1 + 5.14T + 11T^{2} \) |
| 13 | \( 1 + 2.29T + 13T^{2} \) |
| 17 | \( 1 - 1.92T + 17T^{2} \) |
| 19 | \( 1 - 3.15T + 19T^{2} \) |
| 23 | \( 1 - 6.68T + 23T^{2} \) |
| 29 | \( 1 - 7.68T + 29T^{2} \) |
| 31 | \( 1 + 1.20T + 31T^{2} \) |
| 37 | \( 1 + 0.791T + 37T^{2} \) |
| 41 | \( 1 + 2.49T + 41T^{2} \) |
| 43 | \( 1 - 1.15T + 43T^{2} \) |
| 47 | \( 1 + 8.92T + 47T^{2} \) |
| 53 | \( 1 - 7.62T + 53T^{2} \) |
| 59 | \( 1 + 9.24T + 59T^{2} \) |
| 61 | \( 1 - 2.23T + 61T^{2} \) |
| 67 | \( 1 - 14.8T + 67T^{2} \) |
| 71 | \( 1 - 4.46T + 71T^{2} \) |
| 73 | \( 1 + 2.29T + 73T^{2} \) |
| 79 | \( 1 + 9.20T + 79T^{2} \) |
| 83 | \( 1 + 13.9T + 83T^{2} \) |
| 89 | \( 1 + 7.81T + 89T^{2} \) |
| 97 | \( 1 + 4.25T + 97T^{2} \) |
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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
−8.446482033757533056940452304938, −8.159537188143234298063418752883, −7.38716275487410493904879601016, −6.87582965283516068352160110217, −5.67200805944442821460722738715, −4.89448782145485875045772732730, −3.15488960420479644611664137727, −2.46248048932991588836976287666, −1.26018766676849709596599869910, 0,
1.26018766676849709596599869910, 2.46248048932991588836976287666, 3.15488960420479644611664137727, 4.89448782145485875045772732730, 5.67200805944442821460722738715, 6.87582965283516068352160110217, 7.38716275487410493904879601016, 8.159537188143234298063418752883, 8.446482033757533056940452304938
