| L(s) = 1 | − 3-s + 7-s + 9-s + 11-s − 2·13-s − 2·17-s − 4·19-s − 21-s − 4·23-s − 27-s − 6·29-s − 4·31-s − 33-s + 2·37-s + 2·39-s − 6·41-s − 12·43-s + 4·47-s + 49-s + 2·51-s − 6·53-s + 4·57-s + 4·59-s + 2·61-s + 63-s + 12·67-s + 4·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.554·13-s − 0.485·17-s − 0.917·19-s − 0.218·21-s − 0.834·23-s − 0.192·27-s − 1.11·29-s − 0.718·31-s − 0.174·33-s + 0.328·37-s + 0.320·39-s − 0.937·41-s − 1.82·43-s + 0.583·47-s + 1/7·49-s + 0.280·51-s − 0.824·53-s + 0.529·57-s + 0.520·59-s + 0.256·61-s + 0.125·63-s + 1.46·67-s + 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| good | 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−13.49990977315685, −13.12071958974282, −12.56634274794183, −12.25401478388833, −11.58942698270266, −11.32875970047759, −10.91002092624436, −10.29143269086676, −9.862355290054614, −9.494667150868549, −8.739819083561037, −8.411851421146918, −7.898867549755918, −7.146043560336805, −6.962615458329778, −6.328907016836172, −5.766456712836484, −5.355242764913530, −4.758055894025599, −4.238890387950976, −3.824193917072467, −3.106103466919045, −2.285714730628095, −1.827272121086861, −1.266203143841036, 0, 0,
1.266203143841036, 1.827272121086861, 2.285714730628095, 3.106103466919045, 3.824193917072467, 4.238890387950976, 4.758055894025599, 5.355242764913530, 5.766456712836484, 6.328907016836172, 6.962615458329778, 7.146043560336805, 7.898867549755918, 8.411851421146918, 8.739819083561037, 9.494667150868549, 9.862355290054614, 10.29143269086676, 10.91002092624436, 11.32875970047759, 11.58942698270266, 12.25401478388833, 12.56634274794183, 13.12071958974282, 13.49990977315685
