| L(s) = 1 | + (0.104 + 0.994i)2-s + (−1.63 − 0.585i)3-s + (−0.978 + 0.207i)4-s + (0.112 + 0.251i)5-s + (0.412 − 1.68i)6-s + (−2.57 + 0.597i)7-s + (−0.309 − 0.951i)8-s + (2.31 + 1.90i)9-s + (−0.238 + 0.137i)10-s + (0.877 − 3.19i)11-s + (1.71 + 0.233i)12-s + (0.376 + 0.518i)13-s + (−0.863 − 2.50i)14-s + (−0.0352 − 0.475i)15-s + (0.913 − 0.406i)16-s + (0.638 − 6.07i)17-s + ⋯ |
| L(s) = 1 | + (0.0739 + 0.703i)2-s + (−0.941 − 0.338i)3-s + (−0.489 + 0.103i)4-s + (0.0500 + 0.112i)5-s + (0.168 − 0.686i)6-s + (−0.974 + 0.225i)7-s + (−0.109 − 0.336i)8-s + (0.771 + 0.636i)9-s + (−0.0754 + 0.0435i)10-s + (0.264 − 0.964i)11-s + (0.495 + 0.0675i)12-s + (0.104 + 0.143i)13-s + (−0.230 − 0.668i)14-s + (−0.00910 − 0.122i)15-s + (0.228 − 0.101i)16-s + (0.154 − 1.47i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.713 + 0.700i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.713 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.674370 - 0.275487i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.674370 - 0.275487i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.104 - 0.994i)T \) |
| 3 | \( 1 + (1.63 + 0.585i)T \) |
| 7 | \( 1 + (2.57 - 0.597i)T \) |
| 11 | \( 1 + (-0.877 + 3.19i)T \) |
| good | 5 | \( 1 + (-0.112 - 0.251i)T + (-3.34 + 3.71i)T^{2} \) |
| 13 | \( 1 + (-0.376 - 0.518i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.638 + 6.07i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (-0.228 + 1.07i)T + (-17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (-1.05 - 0.608i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.73 + 8.40i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (4.57 + 2.03i)T + (20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (-4.83 - 5.37i)T + (-3.86 + 36.7i)T^{2} \) |
| 41 | \( 1 + (0.939 + 2.89i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 1.12iT - 43T^{2} \) |
| 47 | \( 1 + (0.496 - 2.33i)T + (-42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (3.33 - 7.48i)T + (-35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (1.83 + 8.65i)T + (-53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (1.80 + 4.04i)T + (-40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (5.31 + 9.19i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (6.62 - 9.11i)T + (-21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-2.37 - 11.1i)T + (-66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (-12.3 + 1.29i)T + (77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (3.41 + 2.48i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (15.7 + 9.07i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (8.14 - 5.91i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.08218530598228970345864841588, −9.929105218933532102288134200364, −9.197088106042818832258788010397, −7.999760076426116916043697664363, −6.95626577096031331700397380025, −6.30141069926169073224956903706, −5.54359985075755950400205036032, −4.40151627514292613711414225526, −2.90675575697275070018296307585, −0.53949717027603097291103729771,
1.42751269587881556672406063268, 3.34425018309058149940573488302, 4.27171772685615324893824262016, 5.36719129610794633652447062719, 6.37578361796616445707351685870, 7.28772048514412287189257139067, 8.854305931198840098532638487842, 9.648735130320275986707096072523, 10.44852145003585762956989451628, 10.94070629706199170787130340072
