| L(s) = 1 | − 2·5-s + 2·7-s + 4·11-s − 4·13-s + 2·17-s − 4·23-s − 25-s + 4·31-s − 4·35-s − 2·37-s + 41-s + 12·43-s + 2·47-s − 3·49-s − 4·53-s − 8·55-s − 4·59-s − 10·61-s + 8·65-s + 8·67-s + 10·71-s − 2·73-s + 8·77-s − 14·79-s − 12·83-s − 4·85-s − 10·89-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.755·7-s + 1.20·11-s − 1.10·13-s + 0.485·17-s − 0.834·23-s − 1/5·25-s + 0.718·31-s − 0.676·35-s − 0.328·37-s + 0.156·41-s + 1.82·43-s + 0.291·47-s − 3/7·49-s − 0.549·53-s − 1.07·55-s − 0.520·59-s − 1.28·61-s + 0.992·65-s + 0.977·67-s + 1.18·71-s − 0.234·73-s + 0.911·77-s − 1.57·79-s − 1.31·83-s − 0.433·85-s − 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 41 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−15.59363152031767, −15.26221493711719, −14.49027276047166, −14.19145185576031, −13.89831516467990, −12.73009646283444, −12.41240609279747, −11.83360226425650, −11.52811634096447, −10.95526489291670, −10.21525287572609, −9.625655412733778, −9.133065394246374, −8.340403473479221, −7.910360942530712, −7.409028542963649, −6.837475201462582, −6.051229888002532, −5.448074444888099, −4.495981498165823, −4.315457898800630, −3.547961580225474, −2.730201592583852, −1.863184804045530, −1.066979681280962, 0,
1.066979681280962, 1.863184804045530, 2.730201592583852, 3.547961580225474, 4.315457898800630, 4.495981498165823, 5.448074444888099, 6.051229888002532, 6.837475201462582, 7.409028542963649, 7.910360942530712, 8.340403473479221, 9.133065394246374, 9.625655412733778, 10.21525287572609, 10.95526489291670, 11.52811634096447, 11.83360226425650, 12.41240609279747, 12.73009646283444, 13.89831516467990, 14.19145185576031, 14.49027276047166, 15.26221493711719, 15.59363152031767
