| L(s) = 1 | − 3·2-s + 18·3-s + 5·4-s + 33·5-s − 54·6-s − 99·8-s + 243·9-s − 99·10-s − 1.13e3·11-s + 90·12-s − 925·13-s + 594·15-s − 459·16-s + 324·17-s − 729·18-s − 2.31e3·19-s + 165·20-s + 3.41e3·22-s + 1.59e3·23-s − 1.78e3·24-s − 2.38e3·25-s + 2.77e3·26-s + 2.91e3·27-s − 2.21e3·29-s − 1.78e3·30-s − 4.29e3·31-s + 4.36e3·32-s + ⋯ |
| L(s) = 1 | − 0.530·2-s + 1.15·3-s + 5/32·4-s + 0.590·5-s − 0.612·6-s − 0.546·8-s + 9-s − 0.313·10-s − 2.83·11-s + 0.180·12-s − 1.51·13-s + 0.681·15-s − 0.448·16-s + 0.271·17-s − 0.530·18-s − 1.46·19-s + 0.0922·20-s + 1.50·22-s + 0.629·23-s − 0.631·24-s − 0.762·25-s + 0.805·26-s + 0.769·27-s − 0.489·29-s − 0.361·30-s − 0.802·31-s + 0.753·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + 3 T + p^{2} T^{2} + 3 p^{5} T^{3} + p^{10} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 33 T + 3472 T^{2} - 33 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 1137 T + 645232 T^{2} + 1137 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 925 T + 951450 T^{2} + 925 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 324 T + 1502434 T^{2} - 324 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2311 T + 6241998 T^{2} + 2311 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 1596 T + 7604206 T^{2} - 1596 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2217 T + 42125014 T^{2} + 2217 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4294 T + 25151367 T^{2} + 4294 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 19109 T + 184470078 T^{2} + 19109 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 12858 T + 228491122 T^{2} - 12858 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2771 T + 36114396 T^{2} + 2771 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 23160 T + 570076618 T^{2} - 23160 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 31653 T + 896010526 T^{2} + 31653 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 41097 T + 896994610 T^{2} + 41097 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 42052 T + 1728407262 T^{2} + 42052 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 30763 T + 1698033324 T^{2} - 30763 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 102096 T + 6091648810 T^{2} - 102096 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 28577 T + 2636499108 T^{2} - 28577 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 18464 T + 3332046261 T^{2} + 18464 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 61179 T + 8589312784 T^{2} + 61179 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 29322 T + 11382011794 T^{2} + 29322 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 9791 T + 17134261944 T^{2} - 9791 p^{5} T^{3} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.18531646024113554861615028485, −10.98333057672356782561897038682, −10.75025679854187434642862371755, −10.31637998498592291701011526655, −9.597832961713370677363959296157, −9.460409678122790556063826087154, −8.751786800519158757786823859536, −8.252508069274632845332126271264, −7.65023487465555416172064073946, −7.43894621286793819805019524320, −6.67294266882065078484450509044, −5.80905837720830070576998366880, −5.13406605584572312001463416135, −4.71988752942268134871114721622, −3.61859773359340275618543275722, −2.60443771002624025864218642479, −2.51122278055465735574161802883, −1.82645569143843763439410693965, 0, 0,
1.82645569143843763439410693965, 2.51122278055465735574161802883, 2.60443771002624025864218642479, 3.61859773359340275618543275722, 4.71988752942268134871114721622, 5.13406605584572312001463416135, 5.80905837720830070576998366880, 6.67294266882065078484450509044, 7.43894621286793819805019524320, 7.65023487465555416172064073946, 8.252508069274632845332126271264, 8.751786800519158757786823859536, 9.460409678122790556063826087154, 9.597832961713370677363959296157, 10.31637998498592291701011526655, 10.75025679854187434642862371755, 10.98333057672356782561897038682, 12.18531646024113554861615028485
