| L(s) = 1 | + 3·3-s + 5-s − 7-s + 6·9-s + 3·15-s + 3·17-s − 6·19-s − 3·21-s + 4·23-s − 4·25-s + 9·27-s − 2·29-s + 4·31-s − 35-s − 3·37-s − 5·43-s + 6·45-s + 13·47-s − 6·49-s + 9·51-s + 12·53-s − 18·57-s − 10·59-s + 8·61-s − 6·63-s − 2·67-s + 12·69-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.447·5-s − 0.377·7-s + 2·9-s + 0.774·15-s + 0.727·17-s − 1.37·19-s − 0.654·21-s + 0.834·23-s − 4/5·25-s + 1.73·27-s − 0.371·29-s + 0.718·31-s − 0.169·35-s − 0.493·37-s − 0.762·43-s + 0.894·45-s + 1.89·47-s − 6/7·49-s + 1.26·51-s + 1.64·53-s − 2.38·57-s − 1.30·59-s + 1.02·61-s − 0.755·63-s − 0.244·67-s + 1.44·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 - 13 T + p T^{2} \) | 1.47.an |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
| 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
−13.10770822282579, −12.39979502647275, −12.23480436318638, −11.44876280162251, −10.90573060628516, −10.25453630698018, −10.04646164452676, −9.650500349345723, −9.045645551243914, −8.664846487753547, −8.475489778112041, −7.780454217526994, −7.371149516556782, −6.974862912207411, −6.374874119904012, −5.852551743033755, −5.329483470906718, −4.548159057516187, −4.152707218438118, −3.637505274037987, −3.118494983132771, −2.644078994855116, −2.167224159683966, −1.637120963633684, −1.016014397929237, 0,
1.016014397929237, 1.637120963633684, 2.167224159683966, 2.644078994855116, 3.118494983132771, 3.637505274037987, 4.152707218438118, 4.548159057516187, 5.329483470906718, 5.852551743033755, 6.374874119904012, 6.974862912207411, 7.371149516556782, 7.780454217526994, 8.475489778112041, 8.664846487753547, 9.045645551243914, 9.650500349345723, 10.04646164452676, 10.25453630698018, 10.90573060628516, 11.44876280162251, 12.23480436318638, 12.39979502647275, 13.10770822282579
