| L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 10-s − 2·11-s − 13-s + 16-s − 7·17-s + 4·19-s − 20-s − 2·22-s + 25-s − 26-s + 8·29-s − 10·31-s + 32-s − 7·34-s − 2·37-s + 4·38-s − 40-s + 7·41-s + 2·43-s − 2·44-s + 5·47-s − 7·49-s + 50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s − 0.603·11-s − 0.277·13-s + 1/4·16-s − 1.69·17-s + 0.917·19-s − 0.223·20-s − 0.426·22-s + 1/5·25-s − 0.196·26-s + 1.48·29-s − 1.79·31-s + 0.176·32-s − 1.20·34-s − 0.328·37-s + 0.648·38-s − 0.158·40-s + 1.09·41-s + 0.304·43-s − 0.301·44-s + 0.729·47-s − 49-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 263 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 7 T + p T^{2} \) | 1.41.ah |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 5 T + p T^{2} \) | 1.47.af |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 - 11 T + p T^{2} \) | 1.83.al |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 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
−15.79782020288420, −15.05521419453809, −14.75060696789467, −13.98311255376226, −13.60766905515982, −12.97879526648297, −12.57521999119452, −11.95053124625519, −11.43684938934157, −10.84488019498079, −10.51874071113233, −9.657072520432366, −9.072999758612815, −8.446195845950627, −7.789235515181890, −7.189357592081052, −6.755008093149799, −6.026389840334954, −5.241102531880399, −4.889872197795230, −4.073559357860400, −3.613959951312753, −2.615454173177663, −2.289343369408940, −1.100137574163104, 0,
1.100137574163104, 2.289343369408940, 2.615454173177663, 3.613959951312753, 4.073559357860400, 4.889872197795230, 5.241102531880399, 6.026389840334954, 6.755008093149799, 7.189357592081052, 7.789235515181890, 8.446195845950627, 9.072999758612815, 9.657072520432366, 10.51874071113233, 10.84488019498079, 11.43684938934157, 11.95053124625519, 12.57521999119452, 12.97879526648297, 13.60766905515982, 13.98311255376226, 14.75060696789467, 15.05521419453809, 15.79782020288420
