| L(s) = 1 | − 2-s + 4-s − 4·5-s − 8-s + 4·10-s + 4·11-s + 13-s + 16-s + 2·17-s − 4·19-s − 4·20-s − 4·22-s − 2·23-s + 11·25-s − 26-s − 6·29-s + 4·31-s − 32-s − 2·34-s + 4·37-s + 4·38-s + 4·40-s − 6·41-s − 4·43-s + 4·44-s + 2·46-s − 11·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.78·5-s − 0.353·8-s + 1.26·10-s + 1.20·11-s + 0.277·13-s + 1/4·16-s + 0.485·17-s − 0.917·19-s − 0.894·20-s − 0.852·22-s − 0.417·23-s + 11/5·25-s − 0.196·26-s − 1.11·29-s + 0.718·31-s − 0.176·32-s − 0.342·34-s + 0.657·37-s + 0.648·38-s + 0.632·40-s − 0.937·41-s − 0.609·43-s + 0.603·44-s + 0.294·46-s − 1.55·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 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 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−16.61743605566624, −16.38801658036295, −15.51039757195418, −15.15724082806057, −14.75782125784815, −14.03302874216044, −13.20881119757490, −12.33749737458158, −12.09631953058051, −11.48897793186564, −11.04250440573383, −10.45958448583101, −9.611912109809851, −9.013073470359040, −8.422548321396411, −7.894957881222260, −7.444544656840099, −6.592110048787052, −6.260323858526668, −5.067624991965440, −4.253334438472046, −3.722698486637180, −3.134378131883967, −1.920017963347043, −0.9541523983709955, 0,
0.9541523983709955, 1.920017963347043, 3.134378131883967, 3.722698486637180, 4.253334438472046, 5.067624991965440, 6.260323858526668, 6.592110048787052, 7.444544656840099, 7.894957881222260, 8.422548321396411, 9.013073470359040, 9.611912109809851, 10.45958448583101, 11.04250440573383, 11.48897793186564, 12.09631953058051, 12.33749737458158, 13.20881119757490, 14.03302874216044, 14.75782125784815, 15.15724082806057, 15.51039757195418, 16.38801658036295, 16.61743605566624
