| L(s) = 1 | − 2-s + 2·3-s + 4-s + 5-s − 2·6-s − 8-s + 9-s − 10-s − 11-s + 2·12-s − 2·13-s + 2·15-s + 16-s − 18-s − 19-s + 20-s + 22-s − 2·24-s + 25-s + 2·26-s − 4·27-s − 6·29-s − 2·30-s + 4·31-s − 32-s − 2·33-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.447·5-s − 0.816·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s + 0.577·12-s − 0.554·13-s + 0.516·15-s + 1/4·16-s − 0.235·18-s − 0.229·19-s + 0.223·20-s + 0.213·22-s − 0.408·24-s + 1/5·25-s + 0.392·26-s − 0.769·27-s − 1.11·29-s − 0.365·30-s + 0.718·31-s − 0.176·32-s − 0.348·33-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| 19 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 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.g |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−14.15866005797890, −13.53121889988534, −13.02877879551878, −12.62585258486895, −12.05223177935268, −11.26753474267997, −11.11797317397754, −10.29149437671464, −9.902223298336573, −9.436698638837973, −9.124435748516486, −8.402556500483557, −8.215379617802206, −7.530987181406220, −7.201703717091221, −6.545897058472513, −5.801286816952701, −5.529611788701328, −4.557523583232614, −4.130879719202961, −3.156686138588206, −2.949217802237088, −2.160910750046138, −1.870367141005643, −0.9223034244112003, 0,
0.9223034244112003, 1.870367141005643, 2.160910750046138, 2.949217802237088, 3.156686138588206, 4.130879719202961, 4.557523583232614, 5.529611788701328, 5.801286816952701, 6.545897058472513, 7.201703717091221, 7.530987181406220, 8.215379617802206, 8.402556500483557, 9.124435748516486, 9.436698638837973, 9.902223298336573, 10.29149437671464, 11.11797317397754, 11.26753474267997, 12.05223177935268, 12.62585258486895, 13.02877879551878, 13.53121889988534, 14.15866005797890
