| L(s) = 1 | + 2·2-s + 3·4-s − 4·7-s + 4·8-s + 9-s − 8·14-s + 5·16-s + 2·18-s − 12·23-s − 10·25-s − 12·28-s + 12·29-s + 6·32-s + 3·36-s − 8·37-s − 8·43-s − 24·46-s + 9·49-s − 20·50-s + 12·53-s − 16·56-s + 24·58-s − 4·63-s + 7·64-s + 16·67-s + 4·72-s − 16·74-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 1.51·7-s + 1.41·8-s + 1/3·9-s − 2.13·14-s + 5/4·16-s + 0.471·18-s − 2.50·23-s − 2·25-s − 2.26·28-s + 2.22·29-s + 1.06·32-s + 1/2·36-s − 1.31·37-s − 1.21·43-s − 3.53·46-s + 9/7·49-s − 2.82·50-s + 1.64·53-s − 2.13·56-s + 3.15·58-s − 0.503·63-s + 7/8·64-s + 1.95·67-s + 0.471·72-s − 1.85·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 636804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 636804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−7.973335490421157373200340268575, −7.70397779901611905763392602462, −6.84732233829542762166868447522, −6.67685453158503036830492818980, −6.39461130745147042664210509886, −5.65473838318916067878138758246, −5.61719283222003359694793802526, −4.85679242074249456985197372563, −4.12049989296715617337624484159, −3.89674503164637365373404147869, −3.52845307997211445678021341580, −2.72251484544452322835217288408, −2.32322030951675246311285954744, −1.51207835751420731285626782871, 0,
1.51207835751420731285626782871, 2.32322030951675246311285954744, 2.72251484544452322835217288408, 3.52845307997211445678021341580, 3.89674503164637365373404147869, 4.12049989296715617337624484159, 4.85679242074249456985197372563, 5.61719283222003359694793802526, 5.65473838318916067878138758246, 6.39461130745147042664210509886, 6.67685453158503036830492818980, 6.84732233829542762166868447522, 7.70397779901611905763392602462, 7.973335490421157373200340268575
