| L(s) = 1 | − 2·5-s − 2·7-s + 4·11-s − 6·13-s + 2·17-s + 4·19-s − 8·23-s − 25-s − 8·29-s + 8·31-s + 4·35-s + 2·37-s − 2·41-s + 4·43-s − 3·49-s + 2·53-s − 8·55-s + 4·59-s + 6·61-s + 12·65-s − 8·67-s + 6·73-s − 8·77-s − 4·79-s + 12·83-s − 4·85-s − 2·89-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.755·7-s + 1.20·11-s − 1.66·13-s + 0.485·17-s + 0.917·19-s − 1.66·23-s − 1/5·25-s − 1.48·29-s + 1.43·31-s + 0.676·35-s + 0.328·37-s − 0.312·41-s + 0.609·43-s − 3/7·49-s + 0.274·53-s − 1.07·55-s + 0.520·59-s + 0.768·61-s + 1.48·65-s − 0.977·67-s + 0.702·73-s − 0.911·77-s − 0.450·79-s + 1.31·83-s − 0.433·85-s − 0.211·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.058702766\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.058702766\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 397 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−12.71312381362373, −12.42468472848723, −11.91242716552918, −11.62733764543898, −11.43789321239899, −10.44399016988309, −10.04264971429644, −9.629521359438935, −9.400302813302980, −8.732220906256793, −8.086485616629145, −7.596591758549969, −7.431143694841353, −6.751353298922216, −6.299154522880571, −5.747176219382185, −5.227511377972835, −4.458892320253214, −4.181716793003580, −3.488216533968410, −3.259526377943111, −2.377161046877995, −1.917815716861467, −0.9773133330017247, −0.3302025956572678,
0.3302025956572678, 0.9773133330017247, 1.917815716861467, 2.377161046877995, 3.259526377943111, 3.488216533968410, 4.181716793003580, 4.458892320253214, 5.227511377972835, 5.747176219382185, 6.299154522880571, 6.751353298922216, 7.431143694841353, 7.596591758549969, 8.086485616629145, 8.732220906256793, 9.400302813302980, 9.629521359438935, 10.04264971429644, 10.44399016988309, 11.43789321239899, 11.62733764543898, 11.91242716552918, 12.42468472848723, 12.71312381362373
