| L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 10-s − 11-s − 13-s + 16-s + 5·17-s + 19-s − 20-s − 22-s + 5·23-s − 4·25-s − 26-s + 3·29-s + 2·31-s + 32-s + 5·34-s − 11·37-s + 38-s − 40-s − 43-s − 44-s + 5·46-s + 12·47-s − 4·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s − 0.301·11-s − 0.277·13-s + 1/4·16-s + 1.21·17-s + 0.229·19-s − 0.223·20-s − 0.213·22-s + 1.04·23-s − 4/5·25-s − 0.196·26-s + 0.557·29-s + 0.359·31-s + 0.176·32-s + 0.857·34-s − 1.80·37-s + 0.162·38-s − 0.158·40-s − 0.152·43-s − 0.150·44-s + 0.737·46-s + 1.75·47-s − 0.565·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.092812797\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.092812797\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−16.26265789706071, −15.65975951043402, −15.45160075305888, −14.65773495494849, −14.13113858212283, −13.72651654913888, −12.93782531668437, −12.42488291229198, −11.95870025331733, −11.41248070777691, −10.72380477838025, −10.12847747392878, −9.582683881481982, −8.634519211259682, −8.129456812647844, −7.296586982907935, −7.048058049174671, −6.052690264780199, −5.424745404569252, −4.904040425372583, −4.078549114025484, −3.382655856197344, −2.794875318286483, −1.790649359047577, −0.7294417837706348,
0.7294417837706348, 1.790649359047577, 2.794875318286483, 3.382655856197344, 4.078549114025484, 4.904040425372583, 5.424745404569252, 6.052690264780199, 7.048058049174671, 7.296586982907935, 8.129456812647844, 8.634519211259682, 9.582683881481982, 10.12847747392878, 10.72380477838025, 11.41248070777691, 11.95870025331733, 12.42488291229198, 12.93782531668437, 13.72651654913888, 14.13113858212283, 14.65773495494849, 15.45160075305888, 15.65975951043402, 16.26265789706071
