| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 7-s − 8-s + 9-s + 10-s + 12-s + 5·13-s + 14-s − 15-s + 16-s + 2·17-s − 18-s − 20-s − 21-s + 5·23-s − 24-s − 4·25-s − 5·26-s + 27-s − 28-s + 8·29-s + 30-s − 2·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s + 1.38·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.223·20-s − 0.218·21-s + 1.04·23-s − 0.204·24-s − 4/5·25-s − 0.980·26-s + 0.192·27-s − 0.188·28-s + 1.48·29-s + 0.182·30-s − 0.359·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.862188015\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.862188015\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 19 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + T + p T^{2} \) | 1.59.b |
| 61 | \( 1 - 11 T + p T^{2} \) | 1.61.al |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 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
−15.99928867972156, −15.42951803396648, −15.30529955561836, −14.32288584046715, −13.89246798637964, −13.23824498922334, −12.73592100958144, −11.96554787365097, −11.54223710825058, −10.83347861221070, −10.25567111980846, −9.825619975453337, −8.844282292513775, −8.764227844992817, −8.060110129814281, −7.459920078007158, −6.789673404122352, −6.268273590927155, −5.458094574433879, −4.605402459417623, −3.567871138208159, −3.401649212449343, −2.416461671524998, −1.478666928619803, −0.6819134420982941,
0.6819134420982941, 1.478666928619803, 2.416461671524998, 3.401649212449343, 3.567871138208159, 4.605402459417623, 5.458094574433879, 6.268273590927155, 6.789673404122352, 7.459920078007158, 8.060110129814281, 8.764227844992817, 8.844282292513775, 9.825619975453337, 10.25567111980846, 10.83347861221070, 11.54223710825058, 11.96554787365097, 12.73592100958144, 13.23824498922334, 13.89246798637964, 14.32288584046715, 15.30529955561836, 15.42951803396648, 15.99928867972156
