| L(s) = 1 | + (0.573 + 1.29i)2-s + (−1.48 + 0.888i)3-s + (−1.34 + 1.48i)4-s + (−1.29 − 1.47i)5-s + (−2.00 − 1.41i)6-s + (−0.101 − 0.0134i)7-s + (−2.68 − 0.882i)8-s + (1.42 − 2.64i)9-s + (1.16 − 2.51i)10-s + (−1.90 − 3.86i)11-s + (0.676 − 3.39i)12-s + (1.10 + 3.26i)13-s + (−0.0410 − 0.139i)14-s + (3.23 + 1.04i)15-s + (−0.400 − 3.97i)16-s + (4.05 + 4.05i)17-s + ⋯ |
| L(s) = 1 | + (0.405 + 0.913i)2-s + (−0.858 + 0.513i)3-s + (−0.670 + 0.741i)4-s + (−0.578 − 0.660i)5-s + (−0.817 − 0.576i)6-s + (−0.0384 − 0.00506i)7-s + (−0.950 − 0.312i)8-s + (0.473 − 0.880i)9-s + (0.368 − 0.796i)10-s + (−0.574 − 1.16i)11-s + (0.195 − 0.980i)12-s + (0.307 + 0.905i)13-s + (−0.0109 − 0.0372i)14-s + (0.835 + 0.269i)15-s + (−0.100 − 0.994i)16-s + (0.983 + 0.983i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.346i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 + 0.346i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.708455 - 0.126622i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.708455 - 0.126622i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.573 - 1.29i)T \) |
| 3 | \( 1 + (1.48 - 0.888i)T \) |
| good | 5 | \( 1 + (1.29 + 1.47i)T + (-0.652 + 4.95i)T^{2} \) |
| 7 | \( 1 + (0.101 + 0.0134i)T + (6.76 + 1.81i)T^{2} \) |
| 11 | \( 1 + (1.90 + 3.86i)T + (-6.69 + 8.72i)T^{2} \) |
| 13 | \( 1 + (-1.10 - 3.26i)T + (-10.3 + 7.91i)T^{2} \) |
| 17 | \( 1 + (-4.05 - 4.05i)T + 17iT^{2} \) |
| 19 | \( 1 + (-1.11 + 5.60i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (1.08 + 8.20i)T + (-22.2 + 5.95i)T^{2} \) |
| 29 | \( 1 + (5.22 - 0.342i)T + (28.7 - 3.78i)T^{2} \) |
| 31 | \( 1 + (-1.41 + 0.816i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.60 - 8.09i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (0.445 + 3.38i)T + (-39.6 + 10.6i)T^{2} \) |
| 43 | \( 1 + (2.82 - 1.39i)T + (26.1 - 34.1i)T^{2} \) |
| 47 | \( 1 + (-6.79 + 1.82i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-5.33 + 7.97i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (7.83 + 8.93i)T + (-7.70 + 58.4i)T^{2} \) |
| 61 | \( 1 + (0.769 + 11.7i)T + (-60.4 + 7.96i)T^{2} \) |
| 67 | \( 1 + (5.11 + 2.52i)T + (40.7 + 53.1i)T^{2} \) |
| 71 | \( 1 + (-0.0256 - 0.0619i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-0.411 + 0.992i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-3.77 - 14.0i)T + (-68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (10.3 + 9.07i)T + (10.8 + 82.2i)T^{2} \) |
| 89 | \( 1 + (-6.37 + 2.64i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + (3.64 + 2.10i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.81958263435232106672370811475, −9.670994531299530639754057005140, −8.651671329480008808015286885761, −8.121284098791147211688752699012, −6.78439307587640425850022149940, −6.06683062375474323152165923244, −5.09518525889790253136664333604, −4.35168619911725663931021575567, −3.37178173950061439613505487413, −0.43005418690952541587625285747,
1.43421845960360818830345780160, 2.90107279749756524865825128586, 4.00960223464302723799465135949, 5.33182200854715365510757696874, 5.81765991970805441710580546220, 7.44495032964585018371735182345, 7.66129009106478965120423486812, 9.466197521700695489001209565954, 10.24629361681187474046073573564, 10.85265043771687661741056751694
