| L(s) = 1 | − 3-s − 2·5-s + 7-s + 9-s + 4·11-s − 2·13-s + 2·15-s − 6·17-s − 21-s − 25-s − 27-s + 10·29-s + 8·31-s − 4·33-s − 2·35-s + 2·37-s + 2·39-s − 6·41-s − 4·43-s − 2·45-s + 8·47-s + 49-s + 6·51-s − 6·53-s − 8·55-s − 4·59-s − 10·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.516·15-s − 1.45·17-s − 0.218·21-s − 1/5·25-s − 0.192·27-s + 1.85·29-s + 1.43·31-s − 0.696·33-s − 0.338·35-s + 0.328·37-s + 0.320·39-s − 0.937·41-s − 0.609·43-s − 0.298·45-s + 1.16·47-s + 1/7·49-s + 0.840·51-s − 0.824·53-s − 1.07·55-s − 0.520·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 19 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 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 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−13.85382287913735, −13.39671739794600, −12.57944003531489, −12.18856173420869, −11.91000776633486, −11.35584748885431, −11.16439136987871, −10.40414516227906, −10.02392290487977, −9.397619816210927, −8.839638673608961, −8.326701746956811, −7.997106179181288, −7.127348864115321, −6.943287016960362, −6.265425721404983, −5.995242114975217, −4.880522129151891, −4.681863863889845, −4.263903364269999, −3.640766992264803, −2.910911877287373, −2.230257122958661, −1.450548358423883, −0.7776705325622746, 0,
0.7776705325622746, 1.450548358423883, 2.230257122958661, 2.910911877287373, 3.640766992264803, 4.263903364269999, 4.681863863889845, 4.880522129151891, 5.995242114975217, 6.265425721404983, 6.943287016960362, 7.127348864115321, 7.997106179181288, 8.326701746956811, 8.839638673608961, 9.397619816210927, 10.02392290487977, 10.40414516227906, 11.16439136987871, 11.35584748885431, 11.91000776633486, 12.18856173420869, 12.57944003531489, 13.39671739794600, 13.85382287913735
