| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 2·7-s − 8-s − 2·9-s − 11-s + 12-s + 13-s − 2·14-s + 16-s + 3·17-s + 2·18-s − 7·19-s + 2·21-s + 22-s + 8·23-s − 24-s − 26-s − 5·27-s + 2·28-s − 10·29-s − 8·31-s − 32-s − 33-s − 3·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.755·7-s − 0.353·8-s − 2/3·9-s − 0.301·11-s + 0.288·12-s + 0.277·13-s − 0.534·14-s + 1/4·16-s + 0.727·17-s + 0.471·18-s − 1.60·19-s + 0.436·21-s + 0.213·22-s + 1.66·23-s − 0.204·24-s − 0.196·26-s − 0.962·27-s + 0.377·28-s − 1.85·29-s − 1.43·31-s − 0.176·32-s − 0.174·33-s − 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.476454825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.476454825\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| 43 | \( 1 + T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 7 T + p T^{2} \) | 1.41.ah |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 + T + p T^{2} \) | 1.89.b |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−15.15979255205927, −14.72542902284342, −14.42168069996160, −13.64989912879388, −13.12960048902706, −12.51048237546767, −12.02136216537132, −11.11458460258723, −10.83680922033495, −10.67270534880452, −9.487267577576554, −9.167702841154779, −8.710904069558196, −8.167165764113671, −7.583960060056808, −7.202479410454196, −6.361860960521247, −5.569039042014934, −5.283522939243689, −4.255593230710806, −3.562085125656558, −2.924817221383815, −2.091074275130804, −1.644029862549159, −0.4944540814793941,
0.4944540814793941, 1.644029862549159, 2.091074275130804, 2.924817221383815, 3.562085125656558, 4.255593230710806, 5.283522939243689, 5.569039042014934, 6.361860960521247, 7.202479410454196, 7.583960060056808, 8.167165764113671, 8.710904069558196, 9.167702841154779, 9.487267577576554, 10.67270534880452, 10.83680922033495, 11.11458460258723, 12.02136216537132, 12.51048237546767, 13.12960048902706, 13.64989912879388, 14.42168069996160, 14.72542902284342, 15.15979255205927
