| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 2·7-s − 8-s + 9-s − 10-s + 11-s − 12-s − 2·14-s − 15-s + 16-s − 2·17-s − 18-s + 20-s − 2·21-s − 22-s + 8·23-s + 24-s + 25-s − 27-s + 2·28-s − 8·29-s + 30-s − 6·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s − 0.288·12-s − 0.534·14-s − 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.223·20-s − 0.436·21-s − 0.213·22-s + 1.66·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s + 0.377·28-s − 1.48·29-s + 0.182·30-s − 1.07·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−14.80210118272254, −14.24815168921617, −13.53350625029135, −13.06536012936079, −12.61846662728898, −11.95339147496158, −11.30935326252844, −11.10540487597277, −10.73547251516760, −9.858672236046714, −9.645489712795319, −8.807189479088730, −8.681324744198669, −7.789843872375338, −7.356482725821355, −6.651731220698238, −6.460873780952065, −5.357490053297881, −5.295410957216053, −4.575482187916293, −3.675272772440740, −3.125569503025741, −2.016559687999622, −1.753453471637526, −0.9342699372104728, 0,
0.9342699372104728, 1.753453471637526, 2.016559687999622, 3.125569503025741, 3.675272772440740, 4.575482187916293, 5.295410957216053, 5.357490053297881, 6.460873780952065, 6.651731220698238, 7.356482725821355, 7.789843872375338, 8.681324744198669, 8.807189479088730, 9.645489712795319, 9.858672236046714, 10.73547251516760, 11.10540487597277, 11.30935326252844, 11.95339147496158, 12.61846662728898, 13.06536012936079, 13.53350625029135, 14.24815168921617, 14.80210118272254
