| L(s) = 1 | − 3-s + 2·5-s + 9-s + 4·11-s + 13-s − 2·15-s + 6·17-s + 4·19-s − 25-s − 27-s − 4·29-s − 4·31-s − 4·33-s + 12·37-s − 39-s + 12·41-s + 8·43-s + 2·45-s − 2·47-s − 6·51-s + 8·53-s + 8·55-s − 4·57-s + 4·59-s + 10·61-s + 2·65-s + 14·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s + 1.20·11-s + 0.277·13-s − 0.516·15-s + 1.45·17-s + 0.917·19-s − 1/5·25-s − 0.192·27-s − 0.742·29-s − 0.718·31-s − 0.696·33-s + 1.97·37-s − 0.160·39-s + 1.87·41-s + 1.21·43-s + 0.298·45-s − 0.291·47-s − 0.840·51-s + 1.09·53-s + 1.07·55-s − 0.529·57-s + 0.520·59-s + 1.28·61-s + 0.248·65-s + 1.71·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.344391930\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.344391930\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 12 T + p T^{2} \) | 1.37.am |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−14.86492033658551, −14.50911725211668, −14.20363409977084, −13.51929143248800, −12.87538454622690, −12.60241235591723, −11.71253828925226, −11.52528572040784, −10.92411015466888, −10.14863160975869, −9.683200506768522, −9.376807210286772, −8.782354427950388, −7.771470941829712, −7.495636526152625, −6.709465668454655, −6.080433124687855, −5.605453042939901, −5.353597837689091, −4.143085196693897, −3.928650357270591, −2.944140909475019, −2.162790840006471, −1.242439407922874, −0.8585961873704717,
0.8585961873704717, 1.242439407922874, 2.162790840006471, 2.944140909475019, 3.928650357270591, 4.143085196693897, 5.353597837689091, 5.605453042939901, 6.080433124687855, 6.709465668454655, 7.495636526152625, 7.771470941829712, 8.782354427950388, 9.376807210286772, 9.683200506768522, 10.14863160975869, 10.92411015466888, 11.52528572040784, 11.71253828925226, 12.60241235591723, 12.87538454622690, 13.51929143248800, 14.20363409977084, 14.50911725211668, 14.86492033658551
