| L(s) = 1 | + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s − 5·11-s + 14-s + 16-s − 7·17-s + 19-s + 20-s − 5·22-s − 4·23-s + 25-s + 28-s + 3·29-s − 2·31-s + 32-s − 7·34-s + 35-s + 10·37-s + 38-s + 40-s + 5·41-s − 10·43-s − 5·44-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 − 1.50·11-s + 0.267·14-s + 1/4·16-s − 1.69·17-s + 0.229·19-s + 0.223·20-s − 1.06·22-s − 0.834·23-s + 1/5·25-s + 0.188·28-s + 0.557·29-s − 0.359·31-s + 0.176·32-s − 1.20·34-s + 0.169·35-s + 1.64·37-s + 0.162·38-s + 0.158·40-s + 0.780·41-s − 1.52·43-s − 0.753·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 13 T + p T^{2} \) | 1.47.an |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 3 T + p T^{2} \) | 1.79.d |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−13.86780407887364, −13.36146875094035, −13.10829987708693, −12.66180677232140, −11.98878209133423, −11.54863925311447, −10.99060981269183, −10.58949465391840, −10.19818495082091, −9.578363581629961, −8.927115877070716, −8.483126697091185, −7.773779199206603, −7.525578963079051, −6.805075300359892, −6.216346485632489, −5.824062896240984, −5.211828100747369, −4.712226741801492, −4.301420454812419, −3.612792037569774, −2.730218138520135, −2.404220096285790, −1.928259955059630, −0.9376222693699733, 0,
0.9376222693699733, 1.928259955059630, 2.404220096285790, 2.730218138520135, 3.612792037569774, 4.301420454812419, 4.712226741801492, 5.211828100747369, 5.824062896240984, 6.216346485632489, 6.805075300359892, 7.525578963079051, 7.773779199206603, 8.483126697091185, 8.927115877070716, 9.578363581629961, 10.19818495082091, 10.58949465391840, 10.99060981269183, 11.54863925311447, 11.98878209133423, 12.66180677232140, 13.10829987708693, 13.36146875094035, 13.86780407887364
