| L(s) = 1 | − 5-s + 7-s + 3·11-s − 13-s − 3·17-s − 2·19-s + 3·23-s + 25-s + 3·29-s + 4·31-s − 35-s + 8·37-s + 6·41-s − 5·43-s − 6·49-s − 6·53-s − 3·55-s − 6·59-s + 8·61-s + 65-s − 14·67-s − 6·71-s + 11·73-s + 3·77-s − 79-s − 3·83-s + 3·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s + 0.904·11-s − 0.277·13-s − 0.727·17-s − 0.458·19-s + 0.625·23-s + 1/5·25-s + 0.557·29-s + 0.718·31-s − 0.169·35-s + 1.31·37-s + 0.937·41-s − 0.762·43-s − 6/7·49-s − 0.824·53-s − 0.404·55-s − 0.781·59-s + 1.02·61-s + 0.124·65-s − 1.71·67-s − 0.712·71-s + 1.28·73-s + 0.341·77-s − 0.112·79-s − 0.329·83-s + 0.325·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 79 | \( 1 + T \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
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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
−14.70227880913744, −14.23302950121844, −13.56153232785970, −13.17179802408402, −12.46557958122133, −12.14754538346898, −11.47520107854107, −11.14105395226250, −10.71457159446500, −9.913058539778280, −9.472924296906479, −8.922263046479813, −8.363854059531932, −7.933294160278409, −7.299263372140933, −6.632767344055599, −6.364262593737849, −5.594896423402640, −4.754720351107066, −4.459674481943492, −3.908261684933551, −3.048981647907574, −2.541054818066860, −1.633393921008542, −0.9922200923991103, 0,
0.9922200923991103, 1.633393921008542, 2.541054818066860, 3.048981647907574, 3.908261684933551, 4.459674481943492, 4.754720351107066, 5.594896423402640, 6.364262593737849, 6.632767344055599, 7.299263372140933, 7.933294160278409, 8.363854059531932, 8.922263046479813, 9.472924296906479, 9.913058539778280, 10.71457159446500, 11.14105395226250, 11.47520107854107, 12.14754538346898, 12.46557958122133, 13.17179802408402, 13.56153232785970, 14.23302950121844, 14.70227880913744
