| L(s) = 1 | − 0.540·2-s − 7.70·4-s − 7.47·5-s − 7·7-s + 8.48·8-s + 4.03·10-s + 18.4·11-s + 8.02·13-s + 3.78·14-s + 57.0·16-s − 46.7·17-s − 69.3·19-s + 57.6·20-s − 9.97·22-s + 23·23-s − 69.0·25-s − 4.33·26-s + 53.9·28-s + 107.·29-s + 88.3·31-s − 98.6·32-s + 25.2·34-s + 52.3·35-s + 83.1·37-s + 37.4·38-s − 63.4·40-s − 10.0·41-s + ⋯ |
| L(s) = 1 | − 0.190·2-s − 0.963·4-s − 0.668·5-s − 0.377·7-s + 0.374·8-s + 0.127·10-s + 0.506·11-s + 0.171·13-s + 0.0721·14-s + 0.891·16-s − 0.666·17-s − 0.837·19-s + 0.644·20-s − 0.0966·22-s + 0.208·23-s − 0.552·25-s − 0.0326·26-s + 0.364·28-s + 0.689·29-s + 0.512·31-s − 0.545·32-s + 0.127·34-s + 0.252·35-s + 0.369·37-s + 0.159·38-s − 0.250·40-s − 0.0381·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 23 | \( 1 - 23T \) |
| good | 2 | \( 1 + 0.540T + 8T^{2} \) |
| 5 | \( 1 + 7.47T + 125T^{2} \) |
| 11 | \( 1 - 18.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 8.02T + 2.19e3T^{2} \) |
| 17 | \( 1 + 46.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 69.3T + 6.85e3T^{2} \) |
| 29 | \( 1 - 107.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 88.3T + 2.97e4T^{2} \) |
| 37 | \( 1 - 83.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 10.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 205.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 425.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 578.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 167.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 868.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 173.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 90.2T + 3.57e5T^{2} \) |
| 73 | \( 1 + 620.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.08e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 772.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 202.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 786.T + 9.12e5T^{2} \) |
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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
−8.721438319105380607776180176826, −8.173123194490822363894163522376, −7.21174202724683883963848058927, −6.34562242281798258077138183794, −5.35568883995847384725949158121, −4.19664696997658922541668681147, −3.94055777944456722315456289332, −2.56096328784184429074766171392, −1.01852405616917994607310284277, 0,
1.01852405616917994607310284277, 2.56096328784184429074766171392, 3.94055777944456722315456289332, 4.19664696997658922541668681147, 5.35568883995847384725949158121, 6.34562242281798258077138183794, 7.21174202724683883963848058927, 8.173123194490822363894163522376, 8.721438319105380607776180176826
