| L(s) = 1 | − 3-s − 3·7-s + 9-s + 5·11-s + 13-s − 7·17-s − 4·19-s + 3·21-s − 4·23-s − 27-s + 9·29-s + 5·31-s − 5·33-s − 10·37-s − 39-s + 4·41-s + 2·43-s − 3·47-s + 2·49-s + 7·51-s + 3·53-s + 4·57-s − 59-s + 61-s − 3·63-s − 5·67-s + 4·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.13·7-s + 1/3·9-s + 1.50·11-s + 0.277·13-s − 1.69·17-s − 0.917·19-s + 0.654·21-s − 0.834·23-s − 0.192·27-s + 1.67·29-s + 0.898·31-s − 0.870·33-s − 1.64·37-s − 0.160·39-s + 0.624·41-s + 0.304·43-s − 0.437·47-s + 2/7·49-s + 0.980·51-s + 0.412·53-s + 0.529·57-s − 0.130·59-s + 0.128·61-s − 0.377·63-s − 0.610·67-s + 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7883461742\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7883461742\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + T + p T^{2} \) | 1.59.b |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 15 T + p T^{2} \) | 1.83.p |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−13.61787161670311, −12.88594599847727, −12.58231375051044, −12.07081815117843, −11.65073345360308, −11.17334391869289, −10.59768917486683, −10.11925269152957, −9.754556055884176, −9.070667853566056, −8.670098332251702, −8.374562686380173, −7.395207137688828, −6.743471370891938, −6.542115494968351, −6.265184367276875, −5.657108724427824, −4.788510757856732, −4.221005095062367, −4.035998381034836, −3.215833538685670, −2.583829066381601, −1.855444426048393, −1.169938359782658, −0.2967426254350794,
0.2967426254350794, 1.169938359782658, 1.855444426048393, 2.583829066381601, 3.215833538685670, 4.035998381034836, 4.221005095062367, 4.788510757856732, 5.657108724427824, 6.265184367276875, 6.542115494968351, 6.743471370891938, 7.395207137688828, 8.374562686380173, 8.670098332251702, 9.070667853566056, 9.754556055884176, 10.11925269152957, 10.59768917486683, 11.17334391869289, 11.65073345360308, 12.07081815117843, 12.58231375051044, 12.88594599847727, 13.61787161670311
