| L(s) = 1 | + 2-s + 4-s + 8-s − 5·11-s − 13-s + 16-s + 2·17-s + 2·19-s − 5·22-s − 8·23-s − 26-s + 5·29-s + 4·31-s + 32-s + 2·34-s − 10·37-s + 2·38-s + 9·41-s − 5·43-s − 5·44-s − 8·46-s − 2·47-s − 52-s − 9·53-s + 5·58-s − 9·59-s − 12·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 1.50·11-s − 0.277·13-s + 1/4·16-s + 0.485·17-s + 0.458·19-s − 1.06·22-s − 1.66·23-s − 0.196·26-s + 0.928·29-s + 0.718·31-s + 0.176·32-s + 0.342·34-s − 1.64·37-s + 0.324·38-s + 1.40·41-s − 0.762·43-s − 0.753·44-s − 1.17·46-s − 0.291·47-s − 0.138·52-s − 1.23·53-s + 0.656·58-s − 1.17·59-s − 1.53·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.079848220\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.079848220\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 15 T + p T^{2} \) | 1.83.p |
| 89 | \( 1 - 5 T + p T^{2} \) | 1.89.af |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−12.62331172844721, −12.39639065550978, −11.79092858732474, −11.64818835010743, −10.73348579095507, −10.54203097493512, −10.03281615436918, −9.777643814057508, −8.955357432752059, −8.462044652426808, −7.935423053695826, −7.538700453532763, −7.264262743794713, −6.437337039526357, −5.946251656527341, −5.723026891209074, −4.988519021950121, −4.626710678073166, −4.239971460947219, −3.312654397215305, −3.048340595905143, −2.564263527573796, −1.808041239342342, −1.346386566123985, −0.2294055630620185,
0.2294055630620185, 1.346386566123985, 1.808041239342342, 2.564263527573796, 3.048340595905143, 3.312654397215305, 4.239971460947219, 4.626710678073166, 4.988519021950121, 5.723026891209074, 5.946251656527341, 6.437337039526357, 7.264262743794713, 7.538700453532763, 7.935423053695826, 8.462044652426808, 8.955357432752059, 9.777643814057508, 10.03281615436918, 10.54203097493512, 10.73348579095507, 11.64818835010743, 11.79092858732474, 12.39639065550978, 12.62331172844721
