| L(s) = 1 | + 2-s − 3-s + 4-s + 3·5-s − 6-s + 8-s − 2·9-s + 3·10-s − 12-s + 13-s − 3·15-s + 16-s − 2·18-s + 3·20-s − 24-s − 25-s + 26-s + 5·27-s + 3·29-s − 3·30-s + 16·31-s + 32-s − 2·36-s − 39-s + 3·40-s + 8·43-s − 6·45-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s + 0.353·8-s − 2/3·9-s + 0.948·10-s − 0.288·12-s + 0.277·13-s − 0.774·15-s + 1/4·16-s − 0.471·18-s + 0.670·20-s − 0.204·24-s − 1/5·25-s + 0.196·26-s + 0.962·27-s + 0.557·29-s − 0.547·30-s + 2.87·31-s + 0.176·32-s − 1/3·36-s − 0.160·39-s + 0.474·40-s + 1.21·43-s − 0.894·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133128 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133128 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.562944090\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.562944090\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 43 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−9.492647014700077188749236631226, −8.916534040165162013453685616519, −8.373043863716362399137743946859, −7.943669049816140495804348500308, −7.25037152822252108935275670773, −6.50701942303820210492627086870, −6.27729939922049248607024112623, −5.82386338109679277055337059805, −5.50335222108289563568830040505, −4.65193336146683785242031913289, −4.47014022869735649506910615408, −3.37416346892506264113080318936, −2.74966278588956205441676541830, −2.16201181367036086977867823438, −1.10144646263603310366509369903,
1.10144646263603310366509369903, 2.16201181367036086977867823438, 2.74966278588956205441676541830, 3.37416346892506264113080318936, 4.47014022869735649506910615408, 4.65193336146683785242031913289, 5.50335222108289563568830040505, 5.82386338109679277055337059805, 6.27729939922049248607024112623, 6.50701942303820210492627086870, 7.25037152822252108935275670773, 7.943669049816140495804348500308, 8.373043863716362399137743946859, 8.916534040165162013453685616519, 9.492647014700077188749236631226
