| L(s) = 1 | − 3-s + 3·5-s − 2·7-s + 9-s + 6·11-s − 3·15-s − 3·17-s + 2·19-s + 2·21-s − 6·23-s + 4·25-s − 27-s − 3·29-s + 4·31-s − 6·33-s − 6·35-s − 7·37-s + 3·41-s + 10·43-s + 3·45-s − 6·47-s − 3·49-s + 3·51-s − 3·53-s + 18·55-s − 2·57-s + 7·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s − 0.755·7-s + 1/3·9-s + 1.80·11-s − 0.774·15-s − 0.727·17-s + 0.458·19-s + 0.436·21-s − 1.25·23-s + 4/5·25-s − 0.192·27-s − 0.557·29-s + 0.718·31-s − 1.04·33-s − 1.01·35-s − 1.15·37-s + 0.468·41-s + 1.52·43-s + 0.447·45-s − 0.875·47-s − 3/7·49-s + 0.420·51-s − 0.412·53-s + 2.42·55-s − 0.264·57-s + 0.896·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 13 T + p T^{2} \) | 1.73.an |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−15.45292781790658, −14.58193016028670, −14.11515162375159, −13.82608606012179, −13.22362076585642, −12.60417323527657, −12.22652745667087, −11.55291298975115, −11.13946384124199, −10.38727560375549, −9.735813019451379, −9.638656733924153, −9.015010747185867, −8.452733335070136, −7.485303902210284, −6.754663578143021, −6.495591009823984, −5.905118040565847, −5.576394656460957, −4.609315986511455, −4.057535675578384, −3.381113860705395, −2.464138601280337, −1.734812003991542, −1.163337600518156, 0,
1.163337600518156, 1.734812003991542, 2.464138601280337, 3.381113860705395, 4.057535675578384, 4.609315986511455, 5.576394656460957, 5.905118040565847, 6.495591009823984, 6.754663578143021, 7.485303902210284, 8.452733335070136, 9.015010747185867, 9.638656733924153, 9.735813019451379, 10.38727560375549, 11.13946384124199, 11.55291298975115, 12.22652745667087, 12.60417323527657, 13.22362076585642, 13.82608606012179, 14.11515162375159, 14.58193016028670, 15.45292781790658
