| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 7-s − 8-s + 9-s + 10-s − 11-s + 12-s − 13-s + 14-s − 15-s + 16-s − 18-s − 4·19-s − 20-s − 21-s + 22-s + 9·23-s − 24-s + 25-s + 26-s + 27-s − 28-s − 3·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.218·21-s + 0.213·22-s + 1.87·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s − 0.188·28-s − 0.557·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 317130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.258055929\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.258055929\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 31 | \( 1 \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−12.78049820187655, −12.21529814770738, −11.49111244512626, −11.34390855658567, −10.58031252090553, −10.48547549924494, −9.769368946388256, −9.397887733284621, −8.894283439927114, −8.591873007760324, −8.083852781045421, −7.567712315882514, −7.145315259988035, −6.807991102055135, −6.202473094179253, −5.681100200627122, −4.866161069840810, −4.677907984538547, −3.795637589483088, −3.386439951959238, −2.926340906576468, −2.206928221066223, −1.924565459971429, −0.8485493684875970, −0.5514857963823020,
0.5514857963823020, 0.8485493684875970, 1.924565459971429, 2.206928221066223, 2.926340906576468, 3.386439951959238, 3.795637589483088, 4.677907984538547, 4.866161069840810, 5.681100200627122, 6.202473094179253, 6.807991102055135, 7.145315259988035, 7.567712315882514, 8.083852781045421, 8.591873007760324, 8.894283439927114, 9.397887733284621, 9.769368946388256, 10.48547549924494, 10.58031252090553, 11.34390855658567, 11.49111244512626, 12.21529814770738, 12.78049820187655
