| L(s) = 1 | − 2-s + 4-s − 3·5-s − 8-s + 3·10-s + 11-s − 13-s + 16-s − 3·17-s − 19-s − 3·20-s − 22-s + 23-s + 4·25-s + 26-s − 5·29-s + 6·31-s − 32-s + 3·34-s − 37-s + 38-s + 3·40-s + 3·43-s + 44-s − 46-s + 4·47-s − 4·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.34·5-s − 0.353·8-s + 0.948·10-s + 0.301·11-s − 0.277·13-s + 1/4·16-s − 0.727·17-s − 0.229·19-s − 0.670·20-s − 0.213·22-s + 0.208·23-s + 4/5·25-s + 0.196·26-s − 0.928·29-s + 1.07·31-s − 0.176·32-s + 0.514·34-s − 0.164·37-s + 0.162·38-s + 0.474·40-s + 0.457·43-s + 0.150·44-s − 0.147·46-s + 0.583·47-s − 0.565·50-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 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 3 T + p T^{2} \) | 1.43.ad |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 5 T + p T^{2} \) | 1.73.f |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−16.78323698447264, −16.09399827482698, −15.65931675965976, −15.16315216057467, −14.73905758725540, −13.93073105056007, −13.24430492453363, −12.47434389344524, −12.06239645175693, −11.41003924024125, −11.06390823165249, −10.39374772377931, −9.664357117192378, −9.019148427151036, −8.459415819015386, −7.919575025727473, −7.289438460178257, −6.816442499655881, −6.055420826910087, −5.140588531693467, −4.290250899135209, −3.802783139034837, −2.901217875057725, −2.074961604189141, −0.9161435771988077, 0,
0.9161435771988077, 2.074961604189141, 2.901217875057725, 3.802783139034837, 4.290250899135209, 5.140588531693467, 6.055420826910087, 6.816442499655881, 7.289438460178257, 7.919575025727473, 8.459415819015386, 9.019148427151036, 9.664357117192378, 10.39374772377931, 11.06390823165249, 11.41003924024125, 12.06239645175693, 12.47434389344524, 13.24430492453363, 13.93073105056007, 14.73905758725540, 15.16315216057467, 15.65931675965976, 16.09399827482698, 16.78323698447264
