| L(s) = 1 | + 4·7-s + 11-s − 2·13-s + 6·17-s + 4·19-s − 8·23-s + 10·29-s + 10·37-s + 6·41-s + 8·43-s + 9·49-s − 2·53-s − 12·59-s − 2·61-s + 12·67-s − 8·71-s + 2·73-s + 4·77-s − 8·79-s − 8·83-s + 6·89-s − 8·91-s + 6·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 0.301·11-s − 0.554·13-s + 1.45·17-s + 0.917·19-s − 1.66·23-s + 1.85·29-s + 1.64·37-s + 0.937·41-s + 1.21·43-s + 9/7·49-s − 0.274·53-s − 1.56·59-s − 0.256·61-s + 1.46·67-s − 0.949·71-s + 0.234·73-s + 0.455·77-s − 0.900·79-s − 0.878·83-s + 0.635·89-s − 0.838·91-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.351188413\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.351188413\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 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.ac |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
| 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
−15.80377713409004, −14.97921988320144, −14.41960732344779, −14.08188054083379, −13.91525037234118, −12.74218394898609, −12.32399360072503, −11.72174819086475, −11.48692311731525, −10.66651724691761, −10.09298891940682, −9.637434475996831, −8.917243822678712, −8.143065016708774, −7.706978795729485, −7.506056579416856, −6.363972996584638, −5.836615896929979, −5.184459910824133, −4.529801699838735, −4.062322796221709, −3.040572476857060, −2.369423488694551, −1.434715535612849, −0.8425964472235441,
0.8425964472235441, 1.434715535612849, 2.369423488694551, 3.040572476857060, 4.062322796221709, 4.529801699838735, 5.184459910824133, 5.836615896929979, 6.363972996584638, 7.506056579416856, 7.706978795729485, 8.143065016708774, 8.917243822678712, 9.637434475996831, 10.09298891940682, 10.66651724691761, 11.48692311731525, 11.72174819086475, 12.32399360072503, 12.74218394898609, 13.91525037234118, 14.08188054083379, 14.41960732344779, 14.97921988320144, 15.80377713409004
