| L(s) = 1 | + 3-s + 9-s + 2·11-s − 13-s + 27-s − 8·29-s + 2·33-s + 6·37-s − 39-s + 6·41-s − 4·43-s + 10·47-s − 7·49-s − 2·53-s + 6·59-s + 6·61-s − 4·67-s + 12·71-s − 4·73-s − 12·79-s + 81-s − 4·83-s − 8·87-s − 6·89-s − 4·97-s + 2·99-s + 101-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.603·11-s − 0.277·13-s + 0.192·27-s − 1.48·29-s + 0.348·33-s + 0.986·37-s − 0.160·39-s + 0.937·41-s − 0.609·43-s + 1.45·47-s − 49-s − 0.274·53-s + 0.781·59-s + 0.768·61-s − 0.488·67-s + 1.42·71-s − 0.468·73-s − 1.35·79-s + 1/9·81-s − 0.439·83-s − 0.857·87-s − 0.635·89-s − 0.406·97-s + 0.201·99-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−13.82763423875072, −13.19135080263606, −12.91084600462314, −12.43186418929048, −11.81855267066405, −11.27693327958828, −11.04452940516047, −10.19238755061471, −9.868739768597299, −9.310309831169833, −8.965535709729927, −8.412655820677875, −7.815370204459018, −7.409480264328451, −6.924305297259792, −6.295199373848675, −5.784908495608775, −5.193263172991251, −4.546165700833866, −3.930202386734330, −3.634345446006005, −2.774867012897542, −2.346680369807180, −1.609858749622101, −0.9763523449236468, 0,
0.9763523449236468, 1.609858749622101, 2.346680369807180, 2.774867012897542, 3.634345446006005, 3.930202386734330, 4.546165700833866, 5.193263172991251, 5.784908495608775, 6.295199373848675, 6.924305297259792, 7.409480264328451, 7.815370204459018, 8.412655820677875, 8.965535709729927, 9.310309831169833, 9.868739768597299, 10.19238755061471, 11.04452940516047, 11.27693327958828, 11.81855267066405, 12.43186418929048, 12.91084600462314, 13.19135080263606, 13.82763423875072
