| L(s) = 1 | + 2-s − 3-s + 4-s − 2·5-s − 6-s + 2·7-s + 8-s + 9-s − 2·10-s − 4·11-s − 12-s + 4·13-s + 2·14-s + 2·15-s + 16-s + 2·17-s + 18-s − 2·20-s − 2·21-s − 4·22-s + 4·23-s − 24-s − 25-s + 4·26-s − 27-s + 2·28-s + 2·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 1.20·11-s − 0.288·12-s + 1.10·13-s + 0.534·14-s + 0.516·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.447·20-s − 0.436·21-s − 0.852·22-s + 0.834·23-s − 0.204·24-s − 1/5·25-s + 0.784·26-s − 0.192·27-s + 0.377·28-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206886 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206886 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 - T \) | |
| 3 | \( 1 + T \) | |
| 29 | \( 1 \) | |
| 41 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−13.11522598556205, −12.72525625402016, −12.44206011975628, −11.78267973713429, −11.41518356038642, −11.03964480598079, −10.63217130879213, −10.37138824043915, −9.505187734441698, −8.939795550291437, −8.399001648840771, −7.824767470989485, −7.530894099514736, −7.192159098100974, −6.377570963142960, −5.785531202600025, −5.627933300277725, −4.801566040622054, −4.609758255245221, −3.987972995659913, −3.366538671539108, −2.982603462535958, −2.138153423427298, −1.491444771772199, −0.8365061235271032, 0,
0.8365061235271032, 1.491444771772199, 2.138153423427298, 2.982603462535958, 3.366538671539108, 3.987972995659913, 4.609758255245221, 4.801566040622054, 5.627933300277725, 5.785531202600025, 6.377570963142960, 7.192159098100974, 7.530894099514736, 7.824767470989485, 8.399001648840771, 8.939795550291437, 9.505187734441698, 10.37138824043915, 10.63217130879213, 11.03964480598079, 11.41518356038642, 11.78267973713429, 12.44206011975628, 12.72525625402016, 13.11522598556205
