| L(s) = 1 | − 3.85·2-s − 8.54·3-s + 6.84·4-s + 13.3·5-s + 32.9·6-s − 17.3·7-s + 4.43·8-s + 46.0·9-s − 51.3·10-s − 53.7·11-s − 58.5·12-s + 41.6·13-s + 66.7·14-s − 113.·15-s − 71.8·16-s − 35.6·17-s − 177.·18-s − 19.9·19-s + 91.3·20-s + 148.·21-s + 207.·22-s − 37.8·24-s + 52.8·25-s − 160.·26-s − 162.·27-s − 118.·28-s + 187.·29-s + ⋯ |
| L(s) = 1 | − 1.36·2-s − 1.64·3-s + 0.856·4-s + 1.19·5-s + 2.24·6-s − 0.935·7-s + 0.195·8-s + 1.70·9-s − 1.62·10-s − 1.47·11-s − 1.40·12-s + 0.888·13-s + 1.27·14-s − 1.96·15-s − 1.12·16-s − 0.509·17-s − 2.32·18-s − 0.241·19-s + 1.02·20-s + 1.53·21-s + 2.00·22-s − 0.322·24-s + 0.423·25-s − 1.20·26-s − 1.15·27-s − 0.800·28-s + 1.19·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 23 | \( 1 \) |
| good | 2 | \( 1 + 3.85T + 8T^{2} \) |
| 3 | \( 1 + 8.54T + 27T^{2} \) |
| 5 | \( 1 - 13.3T + 125T^{2} \) |
| 7 | \( 1 + 17.3T + 343T^{2} \) |
| 11 | \( 1 + 53.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 41.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 35.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 19.9T + 6.85e3T^{2} \) |
| 29 | \( 1 - 187.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 2.12T + 2.97e4T^{2} \) |
| 37 | \( 1 - 103.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 476.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 8.51T + 7.95e4T^{2} \) |
| 47 | \( 1 - 239.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 75.5T + 1.48e5T^{2} \) |
| 59 | \( 1 + 208.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 321.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 694.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 90.5T + 3.57e5T^{2} \) |
| 73 | \( 1 - 106.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 570.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 743.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 560.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 561.T + 9.12e5T^{2} \) |
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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
−10.06487102881762277663176848708, −9.450832672419242462960299757413, −8.353731560402274850818199476939, −7.15498444442108207511469899088, −6.26680392588239407075048853049, −5.72514937557919567444258114417, −4.57942993899956908249561326270, −2.46121548628141887783531630290, −1.04367061823522128873582908656, 0,
1.04367061823522128873582908656, 2.46121548628141887783531630290, 4.57942993899956908249561326270, 5.72514937557919567444258114417, 6.26680392588239407075048853049, 7.15498444442108207511469899088, 8.353731560402274850818199476939, 9.450832672419242462960299757413, 10.06487102881762277663176848708
