| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s − 11-s + 5·13-s − 14-s + 16-s − 2·17-s + 19-s + 22-s − 23-s − 5·26-s + 28-s + 9·29-s − 4·31-s − 32-s + 2·34-s − 8·37-s − 38-s + 10·41-s + 4·43-s − 44-s + 46-s − 7·47-s − 6·49-s + 5·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 0.301·11-s + 1.38·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.229·19-s + 0.213·22-s − 0.208·23-s − 0.980·26-s + 0.188·28-s + 1.67·29-s − 0.718·31-s − 0.176·32-s + 0.342·34-s − 1.31·37-s − 0.162·38-s + 1.56·41-s + 0.609·43-s − 0.150·44-s + 0.147·46-s − 1.02·47-s − 6/7·49-s + 0.693·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 19 | \( 1 - T \) | |
| 23 | \( 1 + T \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 9 T + p T^{2} \) | 1.79.j |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
| 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
−13.29543783001234, −12.81263868162375, −12.29477959930720, −11.84698268384863, −11.23217318740262, −10.96328584135951, −10.53121003912112, −10.10429507980177, −9.399256251945811, −9.055833493462983, −8.561843777904864, −8.067518294268555, −7.829354456851343, −7.040647838121596, −6.658433366732854, −6.125734431153754, −5.666740656325367, −5.013319404237538, −4.486460995759860, −3.817916923330189, −3.298123438339284, −2.667326131388612, −2.028343275949645, −1.395154417277562, −0.8715882750536373, 0,
0.8715882750536373, 1.395154417277562, 2.028343275949645, 2.667326131388612, 3.298123438339284, 3.817916923330189, 4.486460995759860, 5.013319404237538, 5.666740656325367, 6.125734431153754, 6.658433366732854, 7.040647838121596, 7.829354456851343, 8.067518294268555, 8.561843777904864, 9.055833493462983, 9.399256251945811, 10.10429507980177, 10.53121003912112, 10.96328584135951, 11.23217318740262, 11.84698268384863, 12.29477959930720, 12.81263868162375, 13.29543783001234
