| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s + 9-s − 11-s + 12-s + 13-s − 14-s + 16-s + 18-s + 4·19-s − 21-s − 22-s − 6·23-s + 24-s − 5·25-s + 26-s + 27-s − 28-s − 5·31-s + 32-s − 33-s + 36-s + 4·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.235·18-s + 0.917·19-s − 0.218·21-s − 0.213·22-s − 1.25·23-s + 0.204·24-s − 25-s + 0.196·26-s + 0.192·27-s − 0.188·28-s − 0.898·31-s + 0.176·32-s − 0.174·33-s + 1/6·36-s + 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| 37 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + T + p T^{2} \) | 1.67.b |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−14.12370638055718, −13.65732584943607, −13.19521926776030, −12.70016803784389, −12.30253050639188, −11.70763317294791, −11.24436434484824, −10.71076602639392, −10.07782486796225, −9.609785424565587, −9.302742169223388, −8.439098461041767, −8.004927574552560, −7.574482179396630, −6.969770608507693, −6.466994963984685, −5.661073200875047, −5.566713543035595, −4.696427646616944, −4.003174293643937, −3.695139736902822, −3.073605785185890, −2.392468799289302, −1.893715698124673, −1.061497858026161, 0,
1.061497858026161, 1.893715698124673, 2.392468799289302, 3.073605785185890, 3.695139736902822, 4.003174293643937, 4.696427646616944, 5.566713543035595, 5.661073200875047, 6.466994963984685, 6.969770608507693, 7.574482179396630, 8.004927574552560, 8.439098461041767, 9.302742169223388, 9.609785424565587, 10.07782486796225, 10.71076602639392, 11.24436434484824, 11.70763317294791, 12.30253050639188, 12.70016803784389, 13.19521926776030, 13.65732584943607, 14.12370638055718
