| L(s) = 1 | − 1.37·2-s − 6.11·4-s + 10.3·5-s − 5.11·7-s + 19.3·8-s − 14.2·10-s + 55.9·11-s + 37.5·13-s + 7.02·14-s + 22.3·16-s + 23.6·17-s + 39.0·19-s − 63.4·20-s − 76.8·22-s + 71.0·23-s − 17.4·25-s − 51.5·26-s + 31.2·28-s − 28.3·29-s + 12.8·31-s − 185.·32-s − 32.4·34-s − 53.0·35-s − 180.·37-s − 53.5·38-s + 200.·40-s − 215.·41-s + ⋯ |
| L(s) = 1 | − 0.485·2-s − 0.764·4-s + 0.927·5-s − 0.276·7-s + 0.856·8-s − 0.450·10-s + 1.53·11-s + 0.801·13-s + 0.134·14-s + 0.349·16-s + 0.337·17-s + 0.471·19-s − 0.709·20-s − 0.744·22-s + 0.644·23-s − 0.139·25-s − 0.389·26-s + 0.211·28-s − 0.181·29-s + 0.0746·31-s − 1.02·32-s − 0.163·34-s − 0.256·35-s − 0.800·37-s − 0.228·38-s + 0.794·40-s − 0.820·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.226533536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.226533536\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 1.37T + 8T^{2} \) |
| 5 | \( 1 - 10.3T + 125T^{2} \) |
| 7 | \( 1 + 5.11T + 343T^{2} \) |
| 11 | \( 1 - 55.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 37.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 23.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 39.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 71.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 28.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 12.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 180.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 215.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 61.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 61.8T + 1.03e5T^{2} \) |
| 53 | \( 1 - 492.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 789.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 521.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 304.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 270.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 925.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.28e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 713.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 404.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 75.0T + 9.12e5T^{2} \) |
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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
−13.82216648837738113365636432674, −13.03334970135467552696887392049, −11.61951874312439217825415844331, −10.18193927761140093705366970316, −9.378543153758098915250812484491, −8.533224367871037452179829857843, −6.82792884324519527943447860093, −5.49455437992386359685377433967, −3.80380878156222303217447378042, −1.31262223242761053609852637369,
1.31262223242761053609852637369, 3.80380878156222303217447378042, 5.49455437992386359685377433967, 6.82792884324519527943447860093, 8.533224367871037452179829857843, 9.378543153758098915250812484491, 10.18193927761140093705366970316, 11.61951874312439217825415844331, 13.03334970135467552696887392049, 13.82216648837738113365636432674
