| L(s) = 1 | + (0.913 + 0.406i)2-s + (−0.978 − 0.207i)3-s + (0.669 + 0.743i)4-s + (0.258 − 2.45i)5-s + (−0.809 − 0.587i)6-s + (−0.613 + 2.57i)7-s + (0.309 + 0.951i)8-s + (0.913 + 0.406i)9-s + (1.23 − 2.14i)10-s + (3.23 + 0.748i)11-s + (−0.499 − 0.866i)12-s + (2.35 − 1.71i)13-s + (−1.60 + 2.10i)14-s + (−0.764 + 2.35i)15-s + (−0.104 + 0.994i)16-s + (7.22 − 3.21i)17-s + ⋯ |
| L(s) = 1 | + (0.645 + 0.287i)2-s + (−0.564 − 0.120i)3-s + (0.334 + 0.371i)4-s + (0.115 − 1.10i)5-s + (−0.330 − 0.239i)6-s + (−0.231 + 0.972i)7-s + (0.109 + 0.336i)8-s + (0.304 + 0.135i)9-s + (0.391 − 0.677i)10-s + (0.974 + 0.225i)11-s + (−0.144 − 0.249i)12-s + (0.653 − 0.474i)13-s + (−0.429 + 0.561i)14-s + (−0.197 + 0.607i)15-s + (−0.0261 + 0.248i)16-s + (1.75 − 0.780i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.85812 + 0.0738164i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.85812 + 0.0738164i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.913 - 0.406i)T \) |
| 3 | \( 1 + (0.978 + 0.207i)T \) |
| 7 | \( 1 + (0.613 - 2.57i)T \) |
| 11 | \( 1 + (-3.23 - 0.748i)T \) |
| good | 5 | \( 1 + (-0.258 + 2.45i)T + (-4.89 - 1.03i)T^{2} \) |
| 13 | \( 1 + (-2.35 + 1.71i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-7.22 + 3.21i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (0.129 - 0.143i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (1.18 + 2.04i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.615 - 1.89i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (0.0661 + 0.629i)T + (-30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (1.95 - 0.414i)T + (33.8 - 15.0i)T^{2} \) |
| 41 | \( 1 + (1.79 + 5.52i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 6.01T + 43T^{2} \) |
| 47 | \( 1 + (6.96 - 7.73i)T + (-4.91 - 46.7i)T^{2} \) |
| 53 | \( 1 + (-1.38 - 13.1i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (-7.68 - 8.53i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (0.680 - 6.47i)T + (-59.6 - 12.6i)T^{2} \) |
| 67 | \( 1 + (-3.46 + 6.00i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.70 - 1.23i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (9.26 + 10.2i)T + (-7.63 + 72.6i)T^{2} \) |
| 79 | \( 1 + (7.98 + 3.55i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (2.36 + 1.71i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (6.29 + 10.9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.26 - 3.82i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.47837375895959585109049471317, −10.14222618666386522493626192165, −9.149742733401820863127659957729, −8.378554042346256408909615932245, −7.22247910027538247480115515749, −6.02089757700651005287880704119, −5.48854148000672493086128441668, −4.54253235344680354887216507632, −3.19573669955140365341273635771, −1.35606171509949059084575128352,
1.44012861868140897756440893291, 3.41522122978277485686560790178, 3.88725248316602207695282437615, 5.38739943585170937140841426271, 6.49144617224095586209113145515, 6.85091588231875981794594585007, 8.160098406336726529478290021344, 9.856971197416241790976509621228, 10.18120270695844637573020909324, 11.24982364236513652110876585083
