| L(s) = 1 | − 2-s + 4-s + 2·5-s − 8-s − 2·10-s + 4·11-s + 3·13-s + 16-s − 5·17-s − 4·19-s + 2·20-s − 4·22-s + 23-s − 25-s − 3·26-s + 8·29-s + 6·31-s − 32-s + 5·34-s + 37-s + 4·38-s − 2·40-s − 12·41-s + 5·43-s + 4·44-s − 46-s + 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.353·8-s − 0.632·10-s + 1.20·11-s + 0.832·13-s + 1/4·16-s − 1.21·17-s − 0.917·19-s + 0.447·20-s − 0.852·22-s + 0.208·23-s − 1/5·25-s − 0.588·26-s + 1.48·29-s + 1.07·31-s − 0.176·32-s + 0.857·34-s + 0.164·37-s + 0.648·38-s − 0.316·40-s − 1.87·41-s + 0.762·43-s + 0.603·44-s − 0.147·46-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32634 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32634 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 37 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 15 T + p T^{2} \) | 1.61.p |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 + 11 T + p T^{2} \) | 1.83.l |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−15.49115911359295, −14.88109742657470, −14.06736507025264, −13.79448676463839, −13.32872692753718, −12.59922963114058, −12.01635493673715, −11.53549715514350, −10.91294901584494, −10.42565088268921, −9.966787990386329, −9.245057517696279, −8.904435786288519, −8.448984782500700, −7.845965245969840, −6.828480364670816, −6.496669958319443, −6.234613310483418, −5.442041128714845, −4.490915467203340, −4.113580044554220, −3.098859498947023, −2.469359532605013, −1.625699222931151, −1.216899366144906, 0,
1.216899366144906, 1.625699222931151, 2.469359532605013, 3.098859498947023, 4.113580044554220, 4.490915467203340, 5.442041128714845, 6.234613310483418, 6.496669958319443, 6.828480364670816, 7.845965245969840, 8.448984782500700, 8.904435786288519, 9.245057517696279, 9.966787990386329, 10.42565088268921, 10.91294901584494, 11.53549715514350, 12.01635493673715, 12.59922963114058, 13.32872692753718, 13.79448676463839, 14.06736507025264, 14.88109742657470, 15.49115911359295
