| L(s) = 1 | − 3-s − 3·5-s − 7-s + 9-s + 4·11-s − 3·13-s + 3·15-s + 21-s + 23-s + 4·25-s − 27-s − 29-s + 2·31-s − 4·33-s + 3·35-s + 5·37-s + 3·39-s + 5·41-s − 7·43-s − 3·45-s + 3·47-s + 49-s − 12·53-s − 12·55-s − 2·59-s + 6·61-s − 63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.832·13-s + 0.774·15-s + 0.218·21-s + 0.208·23-s + 4/5·25-s − 0.192·27-s − 0.185·29-s + 0.359·31-s − 0.696·33-s + 0.507·35-s + 0.821·37-s + 0.480·39-s + 0.780·41-s − 1.06·43-s − 0.447·45-s + 0.437·47-s + 1/7·49-s − 1.64·53-s − 1.61·55-s − 0.260·59-s + 0.768·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 19 T + p T^{2} \) | 1.97.at |
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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
−15.34808482905838, −14.89001907900757, −14.49202558364615, −13.83027266248859, −13.01783317990130, −12.64847847075629, −11.98319614387712, −11.70366756679342, −11.34145362337678, −10.60961905448411, −10.06271053233507, −9.359904807119729, −8.986469353176041, −8.176772951489005, −7.577326896102938, −7.222921829564335, −6.469040001116926, −6.121491040132986, −5.184439797442989, −4.528640465627175, −4.128894261423489, −3.441833932756260, −2.796068391400136, −1.707289884201380, −0.8034189589871678, 0,
0.8034189589871678, 1.707289884201380, 2.796068391400136, 3.441833932756260, 4.128894261423489, 4.528640465627175, 5.184439797442989, 6.121491040132986, 6.469040001116926, 7.222921829564335, 7.577326896102938, 8.176772951489005, 8.986469353176041, 9.359904807119729, 10.06271053233507, 10.60961905448411, 11.34145362337678, 11.70366756679342, 11.98319614387712, 12.64847847075629, 13.01783317990130, 13.83027266248859, 14.49202558364615, 14.89001907900757, 15.34808482905838
