| L(s) = 1 | − 2-s + 4-s − 8-s + 3·13-s + 16-s − 2·17-s + 19-s − 6·23-s − 3·26-s − 8·29-s + 9·31-s − 32-s + 2·34-s + 3·37-s − 38-s − 2·41-s + 5·43-s + 6·46-s − 2·47-s − 7·49-s + 3·52-s + 2·53-s + 8·58-s − 10·59-s − 7·61-s − 9·62-s + 64-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 0.832·13-s + 1/4·16-s − 0.485·17-s + 0.229·19-s − 1.25·23-s − 0.588·26-s − 1.48·29-s + 1.61·31-s − 0.176·32-s + 0.342·34-s + 0.493·37-s − 0.162·38-s − 0.312·41-s + 0.762·43-s + 0.884·46-s − 0.291·47-s − 49-s + 0.416·52-s + 0.274·53-s + 1.05·58-s − 1.30·59-s − 0.896·61-s − 1.14·62-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 5 T + p T^{2} \) | 1.73.f |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−14.82438874791196, −14.03727241180251, −13.75956575046867, −13.16175433172539, −12.56872440495358, −12.06260844535283, −11.40673057667309, −11.16954104878855, −10.51586961551427, −10.01448080760977, −9.472795829933312, −9.029770644181933, −8.418108036773763, −7.875348812311045, −7.579754232980632, −6.697015502165406, −6.254619092393046, −5.854463095318119, −5.038328816189216, −4.348205395548159, −3.716797992424662, −3.091496120467322, −2.277078441054384, −1.700392220342775, −0.8987622880127309, 0,
0.8987622880127309, 1.700392220342775, 2.277078441054384, 3.091496120467322, 3.716797992424662, 4.348205395548159, 5.038328816189216, 5.854463095318119, 6.254619092393046, 6.697015502165406, 7.579754232980632, 7.875348812311045, 8.418108036773763, 9.029770644181933, 9.472795829933312, 10.01448080760977, 10.51586961551427, 11.16954104878855, 11.40673057667309, 12.06260844535283, 12.56872440495358, 13.16175433172539, 13.75956575046867, 14.03727241180251, 14.82438874791196
