| L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s + 10-s + 4·11-s + 4·13-s − 14-s + 16-s − 20-s − 4·22-s − 2·23-s + 25-s − 4·26-s + 28-s − 2·29-s − 6·31-s − 32-s − 35-s + 10·37-s + 40-s + 10·41-s + 4·44-s + 2·46-s + 49-s − 50-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 + 1.20·11-s + 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.223·20-s − 0.852·22-s − 0.417·23-s + 1/5·25-s − 0.784·26-s + 0.188·28-s − 0.371·29-s − 1.07·31-s − 0.176·32-s − 0.169·35-s + 1.64·37-s + 0.158·40-s + 1.56·41-s + 0.603·44-s + 0.294·46-s + 1/7·49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 182070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.429039100\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.429039100\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−12.91636392238810, −12.70690325531343, −12.13548810719192, −11.43952648623404, −11.25019112422170, −11.01355635489741, −10.34570027883180, −9.636702298863064, −9.339848846503928, −8.918563420895389, −8.366605804870309, −7.895112178423671, −7.549116133313378, −6.924572548182869, −6.283148232928437, −6.112909575078885, −5.404290648353936, −4.678177908325276, −4.046298646434041, −3.720104306555224, −3.132265052119524, −2.259010929348898, −1.759369924359983, −1.037192473530312, −0.5953009758222041,
0.5953009758222041, 1.037192473530312, 1.759369924359983, 2.259010929348898, 3.132265052119524, 3.720104306555224, 4.046298646434041, 4.678177908325276, 5.404290648353936, 6.112909575078885, 6.283148232928437, 6.924572548182869, 7.549116133313378, 7.895112178423671, 8.366605804870309, 8.918563420895389, 9.339848846503928, 9.636702298863064, 10.34570027883180, 11.01355635489741, 11.25019112422170, 11.43952648623404, 12.13548810719192, 12.70690325531343, 12.91636392238810
