| L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s + 6·11-s + 14-s + 16-s + 4·19-s − 20-s − 6·22-s + 25-s − 28-s + 4·31-s − 32-s + 35-s + 4·37-s − 4·38-s + 40-s − 6·41-s − 10·43-s + 6·44-s + 6·47-s + 49-s − 50-s + 6·53-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 + 1.80·11-s + 0.267·14-s + 1/4·16-s + 0.917·19-s − 0.223·20-s − 1.27·22-s + 1/5·25-s − 0.188·28-s + 0.718·31-s − 0.176·32-s + 0.169·35-s + 0.657·37-s − 0.648·38-s + 0.158·40-s − 0.937·41-s − 1.52·43-s + 0.904·44-s + 0.875·47-s + 1/7·49-s − 0.141·50-s + 0.824·53-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 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−13.93419013328811, −13.54969618235340, −12.86753888567953, −12.25650682848744, −11.75921026514641, −11.67529608906770, −11.09748613089221, −10.27599006373420, −10.07273721968335, −9.352683609085499, −9.093946273686025, −8.562126713579717, −8.027507416362531, −7.428454765984921, −6.925559760398204, −6.534635591079461, −6.027491674688225, −5.367212149683019, −4.600688973138643, −4.062209737913832, −3.432879611024292, −3.032719782746748, −2.162255143124412, −1.359358930658467, −0.9477748350058855, 0,
0.9477748350058855, 1.359358930658467, 2.162255143124412, 3.032719782746748, 3.432879611024292, 4.062209737913832, 4.600688973138643, 5.367212149683019, 6.027491674688225, 6.534635591079461, 6.925559760398204, 7.428454765984921, 8.027507416362531, 8.562126713579717, 9.093946273686025, 9.352683609085499, 10.07273721968335, 10.27599006373420, 11.09748613089221, 11.67529608906770, 11.75921026514641, 12.25650682848744, 12.86753888567953, 13.54969618235340, 13.93419013328811
