| L(s) = 1 | + 2-s − 3-s − 4-s − 6-s − 3·8-s + 9-s + 2·11-s + 12-s − 4·13-s − 16-s + 18-s + 2·22-s + 16·23-s + 3·24-s − 6·25-s − 4·26-s − 27-s + 5·32-s − 2·33-s − 36-s + 12·37-s + 4·39-s − 2·44-s + 16·46-s + 16·47-s + 48-s + 2·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.06·8-s + 1/3·9-s + 0.603·11-s + 0.288·12-s − 1.10·13-s − 1/4·16-s + 0.235·18-s + 0.426·22-s + 3.33·23-s + 0.612·24-s − 6/5·25-s − 0.784·26-s − 0.192·27-s + 0.883·32-s − 0.348·33-s − 1/6·36-s + 1.97·37-s + 0.640·39-s − 0.301·44-s + 2.35·46-s + 2.33·47-s + 0.144·48-s + 2/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.289836993\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.289836993\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | $C_1$ | \( 1 + T \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.998460188991818552366129510760, −9.522485991677101039441768614921, −8.975261543279013968212899152873, −8.904851262578921543189533962233, −7.75845614267043749557874942232, −7.46995816184461358823952634116, −6.75687248691737052765776352788, −6.30502880333235153369727352923, −5.44733142268678909820409381105, −5.31622482563829752699303340190, −4.42115778606690496889098125190, −4.20407903283329927801590362766, −3.17498211380366678113802704664, −2.53280394883161174744241592337, −0.938050277873206229946250423607,
0.938050277873206229946250423607, 2.53280394883161174744241592337, 3.17498211380366678113802704664, 4.20407903283329927801590362766, 4.42115778606690496889098125190, 5.31622482563829752699303340190, 5.44733142268678909820409381105, 6.30502880333235153369727352923, 6.75687248691737052765776352788, 7.46995816184461358823952634116, 7.75845614267043749557874942232, 8.904851262578921543189533962233, 8.975261543279013968212899152873, 9.522485991677101039441768614921, 9.998460188991818552366129510760
