| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s + 3·11-s + 12-s + 14-s − 15-s + 16-s + 18-s − 20-s + 21-s + 3·22-s − 4·23-s + 24-s − 4·25-s + 27-s + 28-s + 3·29-s − 30-s + 5·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.904·11-s + 0.288·12-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.235·18-s − 0.223·20-s + 0.218·21-s + 0.639·22-s − 0.834·23-s + 0.204·24-s − 4/5·25-s + 0.192·27-s + 0.188·28-s + 0.557·29-s − 0.182·30-s + 0.898·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.635853335\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.635853335\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 7 T + p T^{2} \) | 1.53.h |
| 59 | \( 1 - 7 T + p T^{2} \) | 1.59.ah |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 3 T + p T^{2} \) | 1.73.d |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−16.04333952198767, −15.65080786633533, −15.28834779732876, −14.45948269385479, −14.07784704381998, −13.81609743803123, −12.95409391530071, −12.31424325267418, −11.91702036852432, −11.36403243142219, −10.69887883970261, −9.964994268355514, −9.424258389734910, −8.548969424942731, −8.169034724961734, −7.432283187087783, −6.862824571998920, −6.146243106255362, −5.474834873375972, −4.555093843531266, −4.054267218397522, −3.538003807166380, −2.586528445859215, −1.887978272597886, −0.8737067665214155,
0.8737067665214155, 1.887978272597886, 2.586528445859215, 3.538003807166380, 4.054267218397522, 4.555093843531266, 5.474834873375972, 6.146243106255362, 6.862824571998920, 7.432283187087783, 8.169034724961734, 8.548969424942731, 9.424258389734910, 9.964994268355514, 10.69887883970261, 11.36403243142219, 11.91702036852432, 12.31424325267418, 12.95409391530071, 13.81609743803123, 14.07784704381998, 14.45948269385479, 15.28834779732876, 15.65080786633533, 16.04333952198767
