| L(s) = 1 | − 2-s − 3-s − 4-s + 5-s + 6-s + 3·8-s + 9-s − 10-s + 12-s − 4·13-s − 15-s − 16-s + 4·17-s − 18-s + 8·19-s − 20-s − 3·24-s + 25-s + 4·26-s − 27-s − 4·29-s + 30-s − 5·32-s − 4·34-s − 36-s − 20·37-s − 8·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s − 1.10·13-s − 0.258·15-s − 1/4·16-s + 0.970·17-s − 0.235·18-s + 1.83·19-s − 0.223·20-s − 0.612·24-s + 1/5·25-s + 0.784·26-s − 0.192·27-s − 0.742·29-s + 0.182·30-s − 0.883·32-s − 0.685·34-s − 1/6·36-s − 3.28·37-s − 1.29·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$ | \( 1 - T \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.974679606209178560349496585785, −8.338930923055989708769347256236, −7.66488013441745380243230842523, −7.64526081628142051255949593555, −6.91188357334163951272135994400, −6.60460944329582925741335209174, −5.60340891427122063107744345734, −5.23920392624592057055772361749, −5.14953646488024942329396142879, −4.35067245182195543943285047345, −3.52619813957683492613022702155, −3.07801231282564442559484356330, −1.87907520889614843272292762156, −1.27903918753275535793143985049, 0,
1.27903918753275535793143985049, 1.87907520889614843272292762156, 3.07801231282564442559484356330, 3.52619813957683492613022702155, 4.35067245182195543943285047345, 5.14953646488024942329396142879, 5.23920392624592057055772361749, 5.60340891427122063107744345734, 6.60460944329582925741335209174, 6.91188357334163951272135994400, 7.64526081628142051255949593555, 7.66488013441745380243230842523, 8.338930923055989708769347256236, 8.974679606209178560349496585785
