| L(s) = 1 | − 2-s − 2·3-s − 4-s + 2·6-s + 3·8-s − 3·9-s + 6·11-s + 2·12-s − 16-s + 10·17-s + 3·18-s + 4·19-s − 6·22-s − 6·24-s − 6·25-s + 14·27-s − 5·32-s − 12·33-s − 10·34-s + 3·36-s − 4·38-s − 4·41-s − 16·43-s − 6·44-s + 2·48-s − 5·49-s + 6·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.816·6-s + 1.06·8-s − 9-s + 1.80·11-s + 0.577·12-s − 1/4·16-s + 2.42·17-s + 0.707·18-s + 0.917·19-s − 1.27·22-s − 1.22·24-s − 6/5·25-s + 2.69·27-s − 0.883·32-s − 2.08·33-s − 1.71·34-s + 1/2·36-s − 0.648·38-s − 0.624·41-s − 2.43·43-s − 0.904·44-s + 0.288·48-s − 5/7·49-s + 0.848·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 440896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 83 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
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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
−8.329893308334208368803964563128, −8.118927993931557797577222055683, −7.50704744345289358390695990950, −6.97450299304430271706246705192, −6.43849364286368598220680598083, −6.03908971095412582423394529075, −5.39435567060411867274578529850, −5.31327024180315445602831381888, −4.73425169524415203719187504641, −3.69889287656033475652720645024, −3.65475901244453246589839654026, −2.84692566072845257253775167246, −1.51392803132280425667443874400, −1.10159956738760523747148561916, 0,
1.10159956738760523747148561916, 1.51392803132280425667443874400, 2.84692566072845257253775167246, 3.65475901244453246589839654026, 3.69889287656033475652720645024, 4.73425169524415203719187504641, 5.31327024180315445602831381888, 5.39435567060411867274578529850, 6.03908971095412582423394529075, 6.43849364286368598220680598083, 6.97450299304430271706246705192, 7.50704744345289358390695990950, 8.118927993931557797577222055683, 8.329893308334208368803964563128
