| L(s) = 1 | + 2-s + 4-s + 2·5-s + 3·7-s + 8-s + 2·10-s + 11-s − 6·13-s + 3·14-s + 16-s − 4·17-s − 19-s + 2·20-s + 22-s − 25-s − 6·26-s + 3·28-s + 4·31-s + 32-s − 4·34-s + 6·35-s − 6·37-s − 38-s + 2·40-s − 6·41-s + 9·43-s + 44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 1.13·7-s + 0.353·8-s + 0.632·10-s + 0.301·11-s − 1.66·13-s + 0.801·14-s + 1/4·16-s − 0.970·17-s − 0.229·19-s + 0.447·20-s + 0.213·22-s − 1/5·25-s − 1.17·26-s + 0.566·28-s + 0.718·31-s + 0.176·32-s − 0.685·34-s + 1.01·35-s − 0.986·37-s − 0.162·38-s + 0.316·40-s − 0.937·41-s + 1.37·43-s + 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136242 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 29 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 - 13 T + p T^{2} \) | 1.71.an |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−13.81409347606887, −13.23630124829310, −12.76544329318317, −12.22879731686349, −11.86885807183264, −11.29157514556868, −10.97707326833399, −10.21325790743364, −9.974009252064203, −9.412102251444553, −8.753478788081948, −8.350466523214039, −7.654473712180065, −7.248019350069305, −6.634956380136574, −6.283720969877163, −5.445237497012911, −5.194311217090057, −4.683463638435498, −4.224090681905653, −3.551754211923150, −2.633331781590462, −2.241017411182916, −1.860222802696218, −1.082281749855585, 0,
1.082281749855585, 1.860222802696218, 2.241017411182916, 2.633331781590462, 3.551754211923150, 4.224090681905653, 4.683463638435498, 5.194311217090057, 5.445237497012911, 6.283720969877163, 6.634956380136574, 7.248019350069305, 7.654473712180065, 8.350466523214039, 8.753478788081948, 9.412102251444553, 9.974009252064203, 10.21325790743364, 10.97707326833399, 11.29157514556868, 11.86885807183264, 12.22879731686349, 12.76544329318317, 13.23630124829310, 13.81409347606887
