| 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.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 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.ac |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 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.k |
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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
−16.40674973708585, −16.08822499842077, −15.46383221760888, −15.14901997476643, −14.59578458937276, −13.86887160487428, −13.24738331883139, −12.66070483870140, −12.00015623030106, −11.84877816520693, −11.00776234633832, −10.62748276662139, −9.898246602262518, −9.000287182081379, −8.137710746828434, −7.840204579845715, −7.274463899385791, −6.681381009807273, −5.685612511697786, −4.919507781901723, −4.638499978693022, −3.596886811900995, −3.235385134093634, −2.469073579427738, −1.081315754322866, 0,
1.081315754322866, 2.469073579427738, 3.235385134093634, 3.596886811900995, 4.638499978693022, 4.919507781901723, 5.685612511697786, 6.681381009807273, 7.274463899385791, 7.840204579845715, 8.137710746828434, 9.000287182081379, 9.898246602262518, 10.62748276662139, 11.00776234633832, 11.84877816520693, 12.00015623030106, 12.66070483870140, 13.24738331883139, 13.86887160487428, 14.59578458937276, 15.14901997476643, 15.46383221760888, 16.08822499842077, 16.40674973708585
