| L(s) = 1 | + 2-s − 4-s + 5-s − 7-s − 3·8-s + 10-s − 6·11-s − 13-s − 14-s − 16-s + 4·17-s − 4·19-s − 20-s − 6·22-s + 25-s − 26-s + 28-s + 4·29-s + 5·32-s + 4·34-s − 35-s + 8·37-s − 4·38-s − 3·40-s + 6·41-s + 2·43-s + 6·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.447·5-s − 0.377·7-s − 1.06·8-s + 0.316·10-s − 1.80·11-s − 0.277·13-s − 0.267·14-s − 1/4·16-s + 0.970·17-s − 0.917·19-s − 0.223·20-s − 1.27·22-s + 1/5·25-s − 0.196·26-s + 0.188·28-s + 0.742·29-s + 0.883·32-s + 0.685·34-s − 0.169·35-s + 1.31·37-s − 0.648·38-s − 0.474·40-s + 0.937·41-s + 0.304·43-s + 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4095 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4095 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.658286789\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.658286789\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−8.200109623937110972678281518993, −7.926433559933441013133611029161, −6.75817195816744273833310071561, −5.98566302017282623005442435709, −5.36986192821416057446451689039, −4.79914231490961151265743023397, −3.91384611069842200401537022035, −2.92643279177267796676948883958, −2.37628438383618005401422317867, −0.63534511591295318876592722009,
0.63534511591295318876592722009, 2.37628438383618005401422317867, 2.92643279177267796676948883958, 3.91384611069842200401537022035, 4.79914231490961151265743023397, 5.36986192821416057446451689039, 5.98566302017282623005442435709, 6.75817195816744273833310071561, 7.926433559933441013133611029161, 8.200109623937110972678281518993
