| L(s) = 1 | + 2-s + 4-s + 3·5-s + 7-s + 8-s + 3·10-s + 11-s + 13-s + 14-s + 16-s − 4·17-s − 19-s + 3·20-s + 22-s + 4·23-s + 4·25-s + 26-s + 28-s − 9·29-s + 2·31-s + 32-s − 4·34-s + 3·35-s − 2·37-s − 38-s + 3·40-s + 8·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.34·5-s + 0.377·7-s + 0.353·8-s + 0.948·10-s + 0.301·11-s + 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.970·17-s − 0.229·19-s + 0.670·20-s + 0.213·22-s + 0.834·23-s + 4/5·25-s + 0.196·26-s + 0.188·28-s − 1.67·29-s + 0.359·31-s + 0.176·32-s − 0.685·34-s + 0.507·35-s − 0.328·37-s − 0.162·38-s + 0.474·40-s + 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33282 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33282 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.874770546\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.874770546\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 43 | \( 1 \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 47 | \( 1 - 11 T + p T^{2} \) | 1.47.al |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 17 T + p T^{2} \) | 1.97.ar |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.75113291369737, −14.51430551147520, −13.87158981357139, −13.47051010847770, −12.97515751976057, −12.65667179103914, −11.84204729895379, −11.21734084116013, −10.88003355064879, −10.35926712731923, −9.465100234815788, −9.308378803438512, −8.602878510377189, −7.893678543441607, −7.140158851494025, −6.644364156441647, −6.093844619010090, −5.518915591756123, −5.099734354596693, −4.306513427843230, −3.781670446646678, −2.866342986564645, −2.175657216626500, −1.760014414032533, −0.8105602806587437,
0.8105602806587437, 1.760014414032533, 2.175657216626500, 2.866342986564645, 3.781670446646678, 4.306513427843230, 5.099734354596693, 5.518915591756123, 6.093844619010090, 6.644364156441647, 7.140158851494025, 7.893678543441607, 8.602878510377189, 9.308378803438512, 9.465100234815788, 10.35926712731923, 10.88003355064879, 11.21734084116013, 11.84204729895379, 12.65667179103914, 12.97515751976057, 13.47051010847770, 13.87158981357139, 14.51430551147520, 14.75113291369737
