| L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 2·6-s − 3·7-s + 9-s − 5·11-s + 2·12-s + 6·14-s − 4·16-s − 3·17-s − 2·18-s − 6·19-s − 3·21-s + 10·22-s − 9·23-s + 27-s − 6·28-s + 8·32-s − 5·33-s + 6·34-s + 2·36-s − 3·37-s + 12·38-s − 5·41-s + 6·42-s + 6·43-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 0.816·6-s − 1.13·7-s + 1/3·9-s − 1.50·11-s + 0.577·12-s + 1.60·14-s − 16-s − 0.727·17-s − 0.471·18-s − 1.37·19-s − 0.654·21-s + 2.13·22-s − 1.87·23-s + 0.192·27-s − 1.13·28-s + 1.41·32-s − 0.870·33-s + 1.02·34-s + 1/3·36-s − 0.493·37-s + 1.94·38-s − 0.780·41-s + 0.925·42-s + 0.914·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 2 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 3 T + p T^{2} \) | 1.61.d |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 15 T + p T^{2} \) | 1.79.p |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - T + p T^{2} \) | 1.89.ab |
| 97 | \( 1 - 3 T + p T^{2} \) | 1.97.ad |
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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
−16.79596484656123, −16.32850928431648, −15.79021646658008, −15.55042606492500, −14.78882621866028, −13.94780071544023, −13.31095322793033, −13.06406420249759, −12.35666226846661, −11.57687180162838, −10.72069166664527, −10.20587446598675, −10.10607966694164, −9.247111946748769, −8.731297502916585, −8.216838698105830, −7.701680690048815, −7.033494804637353, −6.398720485834560, −5.734672167515530, −4.615520956538426, −4.011166697544385, −2.964577213664752, −2.352919292510728, −1.671614075766383, 0, 0,
1.671614075766383, 2.352919292510728, 2.964577213664752, 4.011166697544385, 4.615520956538426, 5.734672167515530, 6.398720485834560, 7.033494804637353, 7.701680690048815, 8.216838698105830, 8.731297502916585, 9.247111946748769, 10.10607966694164, 10.20587446598675, 10.72069166664527, 11.57687180162838, 12.35666226846661, 13.06406420249759, 13.31095322793033, 13.94780071544023, 14.78882621866028, 15.55042606492500, 15.79021646658008, 16.32850928431648, 16.79596484656123
