| L(s) = 1 | − 2-s + 18·3-s + 15·4-s − 18·6-s + 18·7-s − 61·8-s + 243·9-s + 242·11-s + 270·12-s + 66·13-s − 18·14-s − 721·16-s + 920·17-s − 243·18-s − 2.93e3·19-s + 324·21-s − 242·22-s − 5.24e3·23-s − 1.09e3·24-s − 66·26-s + 2.91e3·27-s + 270·28-s − 1.26e4·29-s + 9.93e3·31-s + 411·32-s + 4.35e3·33-s − 920·34-s + ⋯ |
| L(s) = 1 | − 0.176·2-s + 1.15·3-s + 0.468·4-s − 0.204·6-s + 0.138·7-s − 0.336·8-s + 9-s + 0.603·11-s + 0.541·12-s + 0.108·13-s − 0.0245·14-s − 0.704·16-s + 0.772·17-s − 0.176·18-s − 1.86·19-s + 0.160·21-s − 0.106·22-s − 2.06·23-s − 0.389·24-s − 0.0191·26-s + 0.769·27-s + 0.0650·28-s − 2.78·29-s + 1.85·31-s + 0.0709·32-s + 0.696·33-s − 0.136·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T - 7 p T^{2} + p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 18 T + 33382 T^{2} - 18 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 66 T - 135542 T^{2} - 66 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 920 T + 3050062 T^{2} - 920 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2932 T + 7100102 T^{2} + 2932 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 5246 T + 19699918 T^{2} + 5246 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 12600 T + 79348870 T^{2} + 12600 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 9936 T + 53013118 T^{2} - 9936 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 5996 T + 139663118 T^{2} + 5996 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 24244 T + 337184038 T^{2} - 24244 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 20360 T + 277332086 T^{2} + 20360 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 5806 T + 419753950 T^{2} - 5806 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 40770 T + 1224329794 T^{2} + 40770 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 18212 T + 455377462 T^{2} - 18212 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 11398 T + 1131625826 T^{2} + 11398 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 65368 T + 3722340342 T^{2} + 65368 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 61446 T + 3799408318 T^{2} - 61446 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 53412 T + 4851340822 T^{2} + 53412 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 17122 T + 5872391094 T^{2} - 17122 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 14304 T + 6955271542 T^{2} - 14304 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 58140 T + 11297620726 T^{2} + 58140 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 183056 T + 25458744990 T^{2} - 183056 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−9.126197029304428619029000857706, −8.987646036458069131085785904062, −8.278329688795916960004225003096, −7.995212754680776481109046564372, −7.82914480601499123186384093658, −7.18784028349530087460768938891, −6.66162487119046635085302755884, −6.24108233882509431498033170524, −6.04000042765675145532494138098, −5.30904514079145725366690301105, −4.45130924396282906482779042956, −4.31185832462644260175739983235, −3.72957877679875271001616437036, −3.25271789498888835424836679121, −2.66250325566398924946354058784, −1.97289590978981145886363757738, −1.87454579579273278149324467103, −1.23613148193189631051358236774, 0, 0,
1.23613148193189631051358236774, 1.87454579579273278149324467103, 1.97289590978981145886363757738, 2.66250325566398924946354058784, 3.25271789498888835424836679121, 3.72957877679875271001616437036, 4.31185832462644260175739983235, 4.45130924396282906482779042956, 5.30904514079145725366690301105, 6.04000042765675145532494138098, 6.24108233882509431498033170524, 6.66162487119046635085302755884, 7.18784028349530087460768938891, 7.82914480601499123186384093658, 7.995212754680776481109046564372, 8.278329688795916960004225003096, 8.987646036458069131085785904062, 9.126197029304428619029000857706
