| L(s) = 1 | − 0.247·2-s + 9·3-s − 31.9·4-s + 8.82·5-s − 2.22·6-s − 254.·7-s + 15.7·8-s + 81·9-s − 2.18·10-s − 561.·11-s − 287.·12-s + 848.·13-s + 62.7·14-s + 79.4·15-s + 1.01e3·16-s + 300.·17-s − 20.0·18-s − 2.66e3·19-s − 281.·20-s − 2.28e3·21-s + 138.·22-s − 4.06e3·23-s + 142.·24-s − 3.04e3·25-s − 209.·26-s + 729·27-s + 8.11e3·28-s + ⋯ |
| L(s) = 1 | − 0.0436·2-s + 0.577·3-s − 0.998·4-s + 0.157·5-s − 0.0252·6-s − 1.95·7-s + 0.0872·8-s + 0.333·9-s − 0.00689·10-s − 1.39·11-s − 0.576·12-s + 1.39·13-s + 0.0855·14-s + 0.0911·15-s + 0.994·16-s + 0.251·17-s − 0.0145·18-s − 1.69·19-s − 0.157·20-s − 1.13·21-s + 0.0611·22-s − 1.60·23-s + 0.0503·24-s − 0.975·25-s − 0.0608·26-s + 0.192·27-s + 1.95·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.6983619148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6983619148\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 157 | \( 1 + 2.46e4T \) |
| good | 2 | \( 1 + 0.247T + 32T^{2} \) |
| 5 | \( 1 - 8.82T + 3.12e3T^{2} \) |
| 7 | \( 1 + 254.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 561.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 848.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 300.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.66e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.06e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.50e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.25e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.15e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.49e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.51e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.23e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.72e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 6.89e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 7.71e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.86e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.21e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.11e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.18e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.03e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.38e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.15e5T + 8.58e9T^{2} \) |
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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
−10.07272834823905445515192579233, −9.355131716887805425469868757511, −8.516897828735649255858402191982, −7.75093057664405774037357131695, −6.31504359724590330480940623310, −5.71187093856853367373779182243, −4.09056759774443755755586562261, −3.49783954118079055042084102254, −2.26308956429854046601034780685, −0.39728120920257106300905731770,
0.39728120920257106300905731770, 2.26308956429854046601034780685, 3.49783954118079055042084102254, 4.09056759774443755755586562261, 5.71187093856853367373779182243, 6.31504359724590330480940623310, 7.75093057664405774037357131695, 8.516897828735649255858402191982, 9.355131716887805425469868757511, 10.07272834823905445515192579233
