| L(s) = 1 | + 7-s − 3·9-s + 2·13-s + 4·17-s − 2·19-s + 5·23-s − 29-s + 2·31-s − 3·37-s − 12·41-s − 11·43-s + 2·47-s + 49-s − 6·53-s + 10·59-s − 4·61-s − 3·63-s + 67-s + 3·71-s − 9·79-s + 9·81-s + 2·83-s − 6·89-s + 2·91-s − 14·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 9-s + 0.554·13-s + 0.970·17-s − 0.458·19-s + 1.04·23-s − 0.185·29-s + 0.359·31-s − 0.493·37-s − 1.87·41-s − 1.67·43-s + 0.291·47-s + 1/7·49-s − 0.824·53-s + 1.30·59-s − 0.512·61-s − 0.377·63-s + 0.122·67-s + 0.356·71-s − 1.01·79-s + 81-s + 0.219·83-s − 0.635·89-s + 0.209·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.565150123\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.565150123\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - T + p T^{2} \) | 1.67.ab |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + 9 T + p T^{2} \) | 1.79.j |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−12.58888355999846, −11.94424280725958, −11.72284681863824, −11.24352823438695, −10.83851859933014, −10.31289973486299, −9.919687490297418, −9.371837567126760, −8.741486486219983, −8.437402850335527, −8.190974953290748, −7.535250644712124, −6.935706456208416, −6.589266372455020, −5.986375961413693, −5.493722040363493, −5.073391564999908, −4.691192543547691, −3.845444300495263, −3.413796338982690, −3.005607392459798, −2.371248169417589, −1.632826641293278, −1.201255897800885, −0.3332733558817247,
0.3332733558817247, 1.201255897800885, 1.632826641293278, 2.371248169417589, 3.005607392459798, 3.413796338982690, 3.845444300495263, 4.691192543547691, 5.073391564999908, 5.493722040363493, 5.986375961413693, 6.589266372455020, 6.935706456208416, 7.535250644712124, 8.190974953290748, 8.437402850335527, 8.741486486219983, 9.371837567126760, 9.919687490297418, 10.31289973486299, 10.83851859933014, 11.24352823438695, 11.72284681863824, 11.94424280725958, 12.58888355999846
