| L(s) = 1 | − 2-s + 3-s + 4-s + 2·5-s − 6-s + 7-s − 8-s + 9-s − 2·10-s + 12-s − 13-s − 14-s + 2·15-s + 16-s − 6·17-s − 18-s + 4·19-s + 2·20-s + 21-s + 8·23-s − 24-s − 25-s + 26-s + 27-s + 28-s − 2·29-s − 2·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.288·12-s − 0.277·13-s − 0.267·14-s + 0.516·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.917·19-s + 0.447·20-s + 0.218·21-s + 1.66·23-s − 0.204·24-s − 1/5·25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s − 0.371·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66066 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66066 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.351556312\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.351556312\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−14.14631181410453, −13.73690572897032, −13.16239613423767, −12.96115823538420, −12.11049472495015, −11.47072097963194, −11.22905863752470, −10.47249267570047, −10.01292496607410, −9.665418966168682, −8.925569240911405, −8.720164139578418, −8.283022016705785, −7.344873790500967, −6.976786127835448, −6.733154152560129, −5.723151651047591, −5.268192021487307, −4.787977282867005, −3.855436452567958, −3.233097699852103, −2.579964235639798, −1.862970246806413, −1.594493294367807, −0.5337485911126474,
0.5337485911126474, 1.594493294367807, 1.862970246806413, 2.579964235639798, 3.233097699852103, 3.855436452567958, 4.787977282867005, 5.268192021487307, 5.723151651047591, 6.733154152560129, 6.976786127835448, 7.344873790500967, 8.283022016705785, 8.720164139578418, 8.925569240911405, 9.665418966168682, 10.01292496607410, 10.47249267570047, 11.22905863752470, 11.47072097963194, 12.11049472495015, 12.96115823538420, 13.16239613423767, 13.73690572897032, 14.14631181410453
