| L(s) = 1 | + 2-s + 4-s − 5-s − 4·7-s + 8-s − 10-s + 11-s − 4·13-s − 4·14-s + 16-s − 6·17-s + 2·19-s − 20-s + 22-s − 6·23-s + 25-s − 4·26-s − 4·28-s − 6·29-s + 8·31-s + 32-s − 6·34-s + 4·35-s + 2·37-s + 2·38-s − 40-s − 6·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.51·7-s + 0.353·8-s − 0.316·10-s + 0.301·11-s − 1.10·13-s − 1.06·14-s + 1/4·16-s − 1.45·17-s + 0.458·19-s − 0.223·20-s + 0.213·22-s − 1.25·23-s + 1/5·25-s − 0.784·26-s − 0.755·28-s − 1.11·29-s + 1.43·31-s + 0.176·32-s − 1.02·34-s + 0.676·35-s + 0.328·37-s + 0.324·38-s − 0.158·40-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 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.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−9.783680798135660213511965945382, −8.790857980472312625483639121388, −7.69116783533421947288205257192, −6.76464464045152724351659373254, −6.31435740331265609071508460642, −5.10562916383792090677967292418, −4.13495771109178706976253108096, −3.28471544996776453968315196115, −2.24881637975690214561478872678, 0,
2.24881637975690214561478872678, 3.28471544996776453968315196115, 4.13495771109178706976253108096, 5.10562916383792090677967292418, 6.31435740331265609071508460642, 6.76464464045152724351659373254, 7.69116783533421947288205257192, 8.790857980472312625483639121388, 9.783680798135660213511965945382
