| L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s + 10-s − 4·11-s − 14-s + 16-s − 7·17-s + 19-s − 20-s + 4·22-s − 3·23-s − 4·25-s + 28-s − 6·29-s − 7·31-s − 32-s + 7·34-s − 35-s + 37-s − 38-s + 40-s − 7·41-s − 5·43-s − 4·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s − 0.353·8-s + 0.316·10-s − 1.20·11-s − 0.267·14-s + 1/4·16-s − 1.69·17-s + 0.229·19-s − 0.223·20-s + 0.852·22-s − 0.625·23-s − 4/5·25-s + 0.188·28-s − 1.11·29-s − 1.25·31-s − 0.176·32-s + 1.20·34-s − 0.169·35-s + 0.164·37-s − 0.162·38-s + 0.158·40-s − 1.09·41-s − 0.762·43-s − 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 13 T + p T^{2} \) | 1.79.n |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 + 7 T + p T^{2} \) | 1.89.h |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−15.91851961682505, −15.34467842042805, −14.89627315719587, −14.34225461725659, −13.47302560652864, −13.11173514927058, −12.68306439004167, −11.72088041337988, −11.37419182546815, −11.08421755822144, −10.26933049181908, −9.917463631625446, −9.200613978663960, −8.570516886701104, −8.133761872902042, −7.654157627192222, −7.007009802776275, −6.534486243043107, −5.546782392318258, −5.241581635667015, −4.306697980853075, −3.743669462999382, −2.865184946971425, −2.103275516122865, −1.598502154876255, 0, 0,
1.598502154876255, 2.103275516122865, 2.865184946971425, 3.743669462999382, 4.306697980853075, 5.241581635667015, 5.546782392318258, 6.534486243043107, 7.007009802776275, 7.654157627192222, 8.133761872902042, 8.570516886701104, 9.200613978663960, 9.917463631625446, 10.26933049181908, 11.08421755822144, 11.37419182546815, 11.72088041337988, 12.68306439004167, 13.11173514927058, 13.47302560652864, 14.34225461725659, 14.89627315719587, 15.34467842042805, 15.91851961682505
