| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s + 6·11-s − 12-s − 13-s + 14-s + 16-s − 2·17-s − 18-s − 4·19-s + 21-s − 6·22-s − 6·23-s + 24-s + 26-s − 27-s − 28-s + 2·29-s + 4·31-s − 32-s − 6·33-s + 2·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 1.80·11-s − 0.288·12-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.917·19-s + 0.218·21-s − 1.27·22-s − 1.25·23-s + 0.204·24-s + 0.196·26-s − 0.192·27-s − 0.188·28-s + 0.371·29-s + 0.718·31-s − 0.176·32-s − 1.04·33-s + 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9034151112\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9034151112\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−16.40683168580879, −15.63681004890194, −15.31213864671528, −14.57539955983377, −13.94536090211081, −13.46389722930170, −12.42812459014998, −12.12284319270819, −11.77410915740855, −10.90751232873290, −10.54117899931850, −9.766141309854813, −9.384195048249520, −8.682570554185729, −8.189616908864090, −7.268606170578295, −6.632868638344075, −6.367827382192306, −5.688115315588352, −4.628112020422357, −4.073730634580445, −3.319124575852807, −2.194141759666467, −1.526446893021416, −0.4976834297724166,
0.4976834297724166, 1.526446893021416, 2.194141759666467, 3.319124575852807, 4.073730634580445, 4.628112020422357, 5.688115315588352, 6.367827382192306, 6.632868638344075, 7.268606170578295, 8.189616908864090, 8.682570554185729, 9.384195048249520, 9.766141309854813, 10.54117899931850, 10.90751232873290, 11.77410915740855, 12.12284319270819, 12.42812459014998, 13.46389722930170, 13.94536090211081, 14.57539955983377, 15.31213864671528, 15.63681004890194, 16.40683168580879
