| L(s) = 1 | − 2-s + 4-s + 2·5-s + 2·7-s − 8-s − 2·10-s − 11-s − 2·14-s + 16-s + 17-s + 5·19-s + 2·20-s + 22-s − 25-s + 2·28-s − 2·31-s − 32-s − 34-s + 4·35-s + 8·37-s − 5·38-s − 2·40-s + 5·41-s − 11·43-s − 44-s + 8·47-s − 3·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.755·7-s − 0.353·8-s − 0.632·10-s − 0.301·11-s − 0.534·14-s + 1/4·16-s + 0.242·17-s + 1.14·19-s + 0.447·20-s + 0.213·22-s − 1/5·25-s + 0.377·28-s − 0.359·31-s − 0.176·32-s − 0.171·34-s + 0.676·35-s + 1.31·37-s − 0.811·38-s − 0.316·40-s + 0.780·41-s − 1.67·43-s − 0.150·44-s + 1.16·47-s − 3/7·49-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 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 11 T + p T^{2} \) | 1.59.l |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 5 T + p T^{2} \) | 1.73.f |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 19 T + p T^{2} \) | 1.97.t |
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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
−15.59809320046603, −14.93034448086934, −14.55263858424209, −13.86293386304336, −13.51229681137162, −12.90207730546889, −12.09603064779968, −11.77784852785757, −11.02141519948155, −10.68304412680053, −9.990907912488514, −9.481995333500228, −9.172023422951274, −8.336844865381169, −7.806486583557322, −7.430422238474198, −6.645904492827517, −5.913870698740339, −5.554817660700399, −4.839943992120599, −4.117970213187441, −3.063702545428003, −2.598480766969772, −1.590708903691650, −1.305791472431542, 0,
1.305791472431542, 1.590708903691650, 2.598480766969772, 3.063702545428003, 4.117970213187441, 4.839943992120599, 5.554817660700399, 5.913870698740339, 6.645904492827517, 7.430422238474198, 7.806486583557322, 8.336844865381169, 9.172023422951274, 9.481995333500228, 9.990907912488514, 10.68304412680053, 11.02141519948155, 11.77784852785757, 12.09603064779968, 12.90207730546889, 13.51229681137162, 13.86293386304336, 14.55263858424209, 14.93034448086934, 15.59809320046603
