| L(s) = 1 | − 2·5-s + 7-s + 3·11-s − 4·13-s + 2·17-s + 8·19-s − 25-s − 5·29-s − 6·31-s − 2·35-s − 4·37-s − 3·41-s − 43-s − 6·47-s − 6·49-s + 4·53-s − 6·55-s − 4·59-s + 8·65-s − 4·67-s − 8·71-s + 10·73-s + 3·77-s − 7·79-s − 5·83-s − 4·85-s + 7·89-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s + 0.904·11-s − 1.10·13-s + 0.485·17-s + 1.83·19-s − 1/5·25-s − 0.928·29-s − 1.07·31-s − 0.338·35-s − 0.657·37-s − 0.468·41-s − 0.152·43-s − 0.875·47-s − 6/7·49-s + 0.549·53-s − 0.809·55-s − 0.520·59-s + 0.992·65-s − 0.488·67-s − 0.949·71-s + 1.17·73-s + 0.341·77-s − 0.787·79-s − 0.548·83-s − 0.433·85-s + 0.741·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 1129 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 7 T + p T^{2} \) | 1.79.h |
| 83 | \( 1 + 5 T + p T^{2} \) | 1.83.f |
| 89 | \( 1 - 7 T + p T^{2} \) | 1.89.ah |
| 97 | \( 1 - 13 T + p T^{2} \) | 1.97.an |
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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
−13.47153529176384, −13.02434443640374, −12.33270339259017, −12.02882716824258, −11.59386416246441, −11.38660473231954, −10.74058273677946, −10.03619431010350, −9.679403668704625, −9.251496150039577, −8.699404066748550, −8.094038189784622, −7.560575028817847, −7.315915644431775, −6.925162267197244, −6.060843559029096, −5.582336095374561, −4.972291533910814, −4.641018917738203, −3.863727492833799, −3.396915374366215, −3.086479027662256, −2.005003295602656, −1.605938389502623, −0.7625826165121010, 0,
0.7625826165121010, 1.605938389502623, 2.005003295602656, 3.086479027662256, 3.396915374366215, 3.863727492833799, 4.641018917738203, 4.972291533910814, 5.582336095374561, 6.060843559029096, 6.925162267197244, 7.315915644431775, 7.560575028817847, 8.094038189784622, 8.699404066748550, 9.251496150039577, 9.679403668704625, 10.03619431010350, 10.74058273677946, 11.38660473231954, 11.59386416246441, 12.02882716824258, 12.33270339259017, 13.02434443640374, 13.47153529176384
