| L(s) = 1 | + 4.46i·2-s − 3.93·4-s − 16.0i·5-s + 72.4·7-s + 53.8i·8-s + 71.5·10-s + 96.1i·11-s + 153.·13-s + 323. i·14-s − 303.·16-s + 72.7i·17-s − 190.·19-s + 63.1i·20-s − 429.·22-s + 14.4i·23-s + ⋯ |
| L(s) = 1 | + 1.11i·2-s − 0.246·4-s − 0.640i·5-s + 1.47·7-s + 0.841i·8-s + 0.715·10-s + 0.794i·11-s + 0.910·13-s + 1.65i·14-s − 1.18·16-s + 0.251i·17-s − 0.528·19-s + 0.157i·20-s − 0.887·22-s + 0.0273i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.46299 + 1.46299i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.46299 + 1.46299i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 4.46iT - 16T^{2} \) |
| 5 | \( 1 + 16.0iT - 625T^{2} \) |
| 7 | \( 1 - 72.4T + 2.40e3T^{2} \) |
| 11 | \( 1 - 96.1iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 153.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 72.7iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 190.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 14.4iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 716. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 302.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 826.T + 1.87e6T^{2} \) |
| 41 | \( 1 + 556. iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 892.T + 3.41e6T^{2} \) |
| 47 | \( 1 + 3.95e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 1.96e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 5.41e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 1.71e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 4.63e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 6.69e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 4.82e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 5.72e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 2.83e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 1.42e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 7.16e3T + 8.85e7T^{2} \) |
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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
−14.31795713133113373345754114147, −12.99214974627739936104216215704, −11.68907315089517778252800620056, −10.70878865523055137692631003357, −8.827883265977777542490786002558, −8.137470994090311484445424306620, −6.98037494688114132909101626386, −5.51971611941578251668075768752, −4.52518872983607294374300415728, −1.72580961234895447035865275240,
1.26106192685789711683003144579, 2.78086085694427877982945905453, 4.28962100489138128727435418502, 6.15187988257844698873668249875, 7.70519031243205350615916905811, 8.991135103277246429746485895862, 10.62127534790562639575547721589, 11.05860353522912133533768530556, 11.87744986234880877521269046106, 13.24978549669333978250807485292
