| L(s) = 1 | − 32·2-s + 162·3-s + 768·4-s − 2.31e3·5-s − 5.18e3·6-s + 3.73e3·7-s − 1.63e4·8-s + 1.96e4·9-s + 7.40e4·10-s + 1.50e4·11-s + 1.24e5·12-s + 5.71e4·13-s − 1.19e5·14-s − 3.74e5·15-s + 3.27e5·16-s − 4.84e5·17-s − 6.29e5·18-s − 6.85e4·19-s − 1.77e6·20-s + 6.05e5·21-s − 4.80e5·22-s − 4.31e5·23-s − 2.65e6·24-s + 1.34e6·25-s − 1.82e6·26-s + 2.12e6·27-s + 2.87e6·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s − 1.65·5-s − 1.63·6-s + 0.588·7-s − 1.41·8-s + 9-s + 2.34·10-s + 0.309·11-s + 1.73·12-s + 0.554·13-s − 0.832·14-s − 1.91·15-s + 5/4·16-s − 1.40·17-s − 1.41·18-s − 0.120·19-s − 2.48·20-s + 0.679·21-s − 0.437·22-s − 0.321·23-s − 1.63·24-s + 0.686·25-s − 0.784·26-s + 0.769·27-s + 0.882·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + p^{4} T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - p^{4} T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - p^{4} T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 2314 T + 32114 p^{3} T^{2} + 2314 p^{9} T^{3} + p^{18} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 534 p T + 11852818 p T^{2} - 534 p^{10} T^{3} + p^{18} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 15020 T + 419343382 T^{2} - 15020 p^{9} T^{3} + p^{18} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 484180 T + 295618325494 T^{2} + 484180 p^{9} T^{3} + p^{18} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 68538 T + 642219890158 T^{2} + 68538 p^{9} T^{3} + p^{18} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 431976 T + 3189924689326 T^{2} + 431976 p^{9} T^{3} + p^{18} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 1864872 T + 1964400640198 T^{2} + 1864872 p^{9} T^{3} + p^{18} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8299954 T + 49199747832990 T^{2} - 8299954 p^{9} T^{3} + p^{18} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 9131400 T + 61778583314710 T^{2} - 9131400 p^{9} T^{3} + p^{18} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 11131186 T + 535822013110330 T^{2} + 11131186 p^{9} T^{3} + p^{18} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 18866232 T + 1050688674155878 T^{2} + 18866232 p^{9} T^{3} + p^{18} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 85216336 T + 3653175155047774 T^{2} + 85216336 p^{9} T^{3} + p^{18} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 153856132 T + 12388358555664622 T^{2} + 153856132 p^{9} T^{3} + p^{18} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 144173624 T + 19239028464157318 T^{2} + 144173624 p^{9} T^{3} + p^{18} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 59443008 T + 914022188337334 T^{2} + 59443008 p^{9} T^{3} + p^{18} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 166548854 T + 51573897202156734 T^{2} + 166548854 p^{9} T^{3} + p^{18} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 15061388 T - 23115651263720018 T^{2} + 15061388 p^{9} T^{3} + p^{18} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 9417608 T + 19941951596546142 T^{2} + 9417608 p^{9} T^{3} + p^{18} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 839937320 T + 413909200502022942 T^{2} + 839937320 p^{9} T^{3} + p^{18} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 699318048 T + 451612178615361238 T^{2} + 699318048 p^{9} T^{3} + p^{18} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 456025130 T + 700638627502586962 T^{2} - 456025130 p^{9} T^{3} + p^{18} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1429586424 T + 1704235462005901102 T^{2} - 1429586424 p^{9} T^{3} + p^{18} T^{4} \) |
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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
−11.88285438318654672660724487483, −11.66690750246370816148884638083, −11.16655365200357594354097215651, −10.69385456385676065266010132856, −9.778131687861258099697008710543, −9.474688155985932296916650904218, −8.574426353822105208231083551620, −8.413589837211907911686011685925, −7.83814457472610941962957315536, −7.64301859820724922477637115106, −6.61602891824790078042385517181, −6.35768279052953653401815202474, −4.67690953548421880162400296571, −4.31567294027852598229279986548, −3.32445652272660540343592839008, −2.92528648570127623628235811743, −1.69565793131419739956162422808, −1.46494253786047836419387462334, 0, 0,
1.46494253786047836419387462334, 1.69565793131419739956162422808, 2.92528648570127623628235811743, 3.32445652272660540343592839008, 4.31567294027852598229279986548, 4.67690953548421880162400296571, 6.35768279052953653401815202474, 6.61602891824790078042385517181, 7.64301859820724922477637115106, 7.83814457472610941962957315536, 8.413589837211907911686011685925, 8.574426353822105208231083551620, 9.474688155985932296916650904218, 9.778131687861258099697008710543, 10.69385456385676065266010132856, 11.16655365200357594354097215651, 11.66690750246370816148884638083, 11.88285438318654672660724487483
