| L(s) = 1 | + 3-s + 9-s − 11-s − 2·13-s + 2·17-s + 4·19-s − 8·23-s + 27-s − 29-s − 4·31-s − 33-s − 2·37-s − 2·39-s − 2·41-s + 4·43-s − 4·47-s − 7·49-s + 2·51-s − 6·53-s + 4·57-s − 4·59-s + 2·61-s − 4·67-s − 8·69-s + 16·71-s − 6·73-s + 8·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.301·11-s − 0.554·13-s + 0.485·17-s + 0.917·19-s − 1.66·23-s + 0.192·27-s − 0.185·29-s − 0.718·31-s − 0.174·33-s − 0.328·37-s − 0.320·39-s − 0.312·41-s + 0.609·43-s − 0.583·47-s − 49-s + 0.280·51-s − 0.824·53-s + 0.529·57-s − 0.520·59-s + 0.256·61-s − 0.488·67-s − 0.963·69-s + 1.89·71-s − 0.702·73-s + 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 382800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 382800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| 29 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−12.86460111677675, −12.19792376441452, −11.92325351651308, −11.34417234838694, −10.90800988819043, −10.30210543274910, −9.924155040237982, −9.521489216854097, −9.247067201001976, −8.461972959356237, −8.134562672600960, −7.704586626385910, −7.344157794824510, −6.787298888085921, −6.241238487611060, −5.644581816196963, −5.300928880340519, −4.675105016563897, −4.203733996541660, −3.560970834408763, −3.194409490485590, −2.654461760786369, −1.886499749204230, −1.679036738699500, −0.7232013481644255, 0,
0.7232013481644255, 1.679036738699500, 1.886499749204230, 2.654461760786369, 3.194409490485590, 3.560970834408763, 4.203733996541660, 4.675105016563897, 5.300928880340519, 5.644581816196963, 6.241238487611060, 6.787298888085921, 7.344157794824510, 7.704586626385910, 8.134562672600960, 8.461972959356237, 9.247067201001976, 9.521489216854097, 9.924155040237982, 10.30210543274910, 10.90800988819043, 11.34417234838694, 11.92325351651308, 12.19792376441452, 12.86460111677675
