| L(s) = 1 | − 3-s + 5-s − 2·7-s + 9-s − 4·11-s − 6·13-s − 15-s + 4·17-s + 2·21-s + 4·23-s + 25-s − 27-s + 6·29-s + 6·31-s + 4·33-s − 2·35-s + 10·37-s + 6·39-s − 4·41-s − 12·43-s + 45-s + 4·47-s − 3·49-s − 4·51-s − 10·53-s − 4·55-s + 10·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s − 1.20·11-s − 1.66·13-s − 0.258·15-s + 0.970·17-s + 0.436·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 1.07·31-s + 0.696·33-s − 0.338·35-s + 1.64·37-s + 0.960·39-s − 0.624·41-s − 1.82·43-s + 0.149·45-s + 0.583·47-s − 3/7·49-s − 0.560·51-s − 1.37·53-s − 0.539·55-s + 1.30·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 346560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 346560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 19 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.91583548418583, −12.29847054229544, −11.97284300110833, −11.47439312075936, −10.97633266742065, −10.23646422310585, −10.07553378411430, −9.837111145981220, −9.399902190068344, −8.582496815648084, −8.191658086649603, −7.640534877424014, −7.195675515079240, −6.766806980944660, −6.202858037715484, −5.848432830675622, −5.144342741143721, −4.850829571821297, −4.606624275520017, −3.592900259615766, −3.023790450210429, −2.685274379425632, −2.172976603395242, −1.287335396879461, −0.6524532793963245, 0,
0.6524532793963245, 1.287335396879461, 2.172976603395242, 2.685274379425632, 3.023790450210429, 3.592900259615766, 4.606624275520017, 4.850829571821297, 5.144342741143721, 5.848432830675622, 6.202858037715484, 6.766806980944660, 7.195675515079240, 7.640534877424014, 8.191658086649603, 8.582496815648084, 9.399902190068344, 9.837111145981220, 10.07553378411430, 10.23646422310585, 10.97633266742065, 11.47439312075936, 11.97284300110833, 12.29847054229544, 12.91583548418583
