| L(s) = 1 | − 2-s − 4-s + 5-s + 3·8-s − 10-s + 2·11-s − 2·13-s − 16-s − 6·19-s − 20-s − 2·22-s + 25-s + 2·26-s − 10·29-s − 8·31-s − 5·32-s − 4·37-s + 6·38-s + 3·40-s − 10·41-s − 43-s − 2·44-s − 50-s + 2·52-s − 12·53-s + 2·55-s + 10·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.447·5-s + 1.06·8-s − 0.316·10-s + 0.603·11-s − 0.554·13-s − 1/4·16-s − 1.37·19-s − 0.223·20-s − 0.426·22-s + 1/5·25-s + 0.392·26-s − 1.85·29-s − 1.43·31-s − 0.883·32-s − 0.657·37-s + 0.973·38-s + 0.474·40-s − 1.56·41-s − 0.152·43-s − 0.301·44-s − 0.141·50-s + 0.277·52-s − 1.64·53-s + 0.269·55-s + 1.31·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 94815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 43 | \( 1 + T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−14.22096486431936, −13.95557662881727, −13.20694895606434, −12.72691821903453, −12.66909998067566, −11.74933359724052, −11.12130227113126, −10.87504608874639, −10.19777317604021, −9.692293896595705, −9.448818258607802, −8.785081081313354, −8.505715829697075, −7.900288595999572, −7.230219215400315, −6.882696147639809, −6.254851194660994, −5.434433877696908, −5.218101794989778, −4.407523695145440, −3.911835885211577, −3.395765091297545, −2.396396958312733, −1.731696122776196, −1.409285506797384, 0, 0,
1.409285506797384, 1.731696122776196, 2.396396958312733, 3.395765091297545, 3.911835885211577, 4.407523695145440, 5.218101794989778, 5.434433877696908, 6.254851194660994, 6.882696147639809, 7.230219215400315, 7.900288595999572, 8.505715829697075, 8.785081081313354, 9.448818258607802, 9.692293896595705, 10.19777317604021, 10.87504608874639, 11.12130227113126, 11.74933359724052, 12.66909998067566, 12.72691821903453, 13.20694895606434, 13.95557662881727, 14.22096486431936
