| L(s) = 1 | + 2.75·2-s − 24.4·4-s − 5.39·5-s − 153.·7-s − 155.·8-s − 14.8·10-s − 761.·11-s + 217.·13-s − 423.·14-s + 352.·16-s + 1.05e3·17-s − 2.36e3·19-s + 131.·20-s − 2.09e3·22-s − 3.57e3·23-s − 3.09e3·25-s + 600.·26-s + 3.74e3·28-s − 538.·29-s − 6.08e3·31-s + 5.94e3·32-s + 2.92e3·34-s + 828.·35-s − 1.26e4·37-s − 6.52e3·38-s + 838.·40-s + 3.78e3·41-s + ⋯ |
| L(s) = 1 | + 0.487·2-s − 0.762·4-s − 0.0965·5-s − 1.18·7-s − 0.858·8-s − 0.0470·10-s − 1.89·11-s + 0.357·13-s − 0.576·14-s + 0.343·16-s + 0.888·17-s − 1.50·19-s + 0.0735·20-s − 0.924·22-s − 1.40·23-s − 0.990·25-s + 0.174·26-s + 0.902·28-s − 0.118·29-s − 1.13·31-s + 1.02·32-s + 0.433·34-s + 0.114·35-s − 1.51·37-s − 0.733·38-s + 0.0828·40-s + 0.351·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.3492222040\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3492222040\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 + 3.48e3T \) |
| good | 2 | \( 1 - 2.75T + 32T^{2} \) |
| 5 | \( 1 + 5.39T + 3.12e3T^{2} \) |
| 7 | \( 1 + 153.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 761.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 217.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.05e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.36e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.57e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 538.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.08e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.26e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.78e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.07e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.97e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.80e4T + 4.18e8T^{2} \) |
| 61 | \( 1 + 8.69e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.66e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.66e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.79e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.47e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.80e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.77e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.47e4T + 8.58e9T^{2} \) |
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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
−10.15768051432737476191374506593, −9.204354464936766440522884918706, −8.281339443088461949767703697204, −7.42577847734396677518216665344, −5.96491822756424814723615630227, −5.58149391688767602956407619583, −4.23633792187136736611628828143, −3.44155120751591128120128807953, −2.33012176120365431459547127405, −0.25556958192687723602350187279,
0.25556958192687723602350187279, 2.33012176120365431459547127405, 3.44155120751591128120128807953, 4.23633792187136736611628828143, 5.58149391688767602956407619583, 5.96491822756424814723615630227, 7.42577847734396677518216665344, 8.281339443088461949767703697204, 9.204354464936766440522884918706, 10.15768051432737476191374506593
