| L(s) = 1 | + (0.652 − 1.25i)2-s + (−0.116 + 0.280i)3-s + (−1.14 − 1.63i)4-s + (0.922 − 1.80i)5-s + (0.276 + 0.329i)6-s + (−0.274 − 1.14i)7-s + (−2.80 + 0.370i)8-s + (2.05 + 2.05i)9-s + (−1.66 − 2.33i)10-s + (−0.0847 + 0.0992i)11-s + (0.593 − 0.131i)12-s + (−2.37 − 3.87i)13-s + (−1.61 − 0.401i)14-s + (0.400 + 0.469i)15-s + (−1.36 + 3.75i)16-s + (0.0717 + 0.911i)17-s + ⋯ |
| L(s) = 1 | + (0.461 − 0.887i)2-s + (−0.0671 + 0.162i)3-s + (−0.573 − 0.818i)4-s + (0.412 − 0.809i)5-s + (0.112 + 0.134i)6-s + (−0.103 − 0.431i)7-s + (−0.991 + 0.130i)8-s + (0.685 + 0.685i)9-s + (−0.527 − 0.739i)10-s + (−0.0255 + 0.0299i)11-s + (0.171 − 0.0380i)12-s + (−0.658 − 1.07i)13-s + (−0.430 − 0.107i)14-s + (0.103 + 0.121i)15-s + (−0.341 + 0.939i)16-s + (0.0174 + 0.221i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.123 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.123 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.911444 - 1.03197i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.911444 - 1.03197i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.652 + 1.25i)T \) |
| 41 | \( 1 + (6.31 - 1.06i)T \) |
| good | 3 | \( 1 + (0.116 - 0.280i)T + (-2.12 - 2.12i)T^{2} \) |
| 5 | \( 1 + (-0.922 + 1.80i)T + (-2.93 - 4.04i)T^{2} \) |
| 7 | \( 1 + (0.274 + 1.14i)T + (-6.23 + 3.17i)T^{2} \) |
| 11 | \( 1 + (0.0847 - 0.0992i)T + (-1.72 - 10.8i)T^{2} \) |
| 13 | \( 1 + (2.37 + 3.87i)T + (-5.90 + 11.5i)T^{2} \) |
| 17 | \( 1 + (-0.0717 - 0.911i)T + (-16.7 + 2.65i)T^{2} \) |
| 19 | \( 1 + (-6.42 - 3.93i)T + (8.62 + 16.9i)T^{2} \) |
| 23 | \( 1 + (-1.18 - 0.861i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.0293 + 0.373i)T + (-28.6 - 4.53i)T^{2} \) |
| 31 | \( 1 + (2.86 - 8.80i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.33 - 4.12i)T + (-29.9 + 21.7i)T^{2} \) |
| 43 | \( 1 + (0.623 + 3.93i)T + (-40.8 + 13.2i)T^{2} \) |
| 47 | \( 1 + (-1.80 + 7.50i)T + (-41.8 - 21.3i)T^{2} \) |
| 53 | \( 1 + (3.50 + 0.276i)T + (52.3 + 8.29i)T^{2} \) |
| 59 | \( 1 + (-2.06 + 2.83i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-0.586 + 3.70i)T + (-58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (5.36 - 4.58i)T + (10.4 - 66.1i)T^{2} \) |
| 71 | \( 1 + (6.33 + 5.40i)T + (11.1 + 70.1i)T^{2} \) |
| 73 | \( 1 + (6.90 - 6.90i)T - 73iT^{2} \) |
| 79 | \( 1 + (12.4 + 5.14i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + 15.8iT - 83T^{2} \) |
| 89 | \( 1 + (5.88 - 1.41i)T + (79.2 - 40.4i)T^{2} \) |
| 97 | \( 1 + (-1.29 + 1.10i)T + (15.1 - 95.8i)T^{2} \) |
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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
−12.69083836112557298240042834324, −11.73292902608550261841244714030, −10.30280307005648563593139933550, −10.04522416590311173316378091647, −8.744973192440989079390021927802, −7.36525462050641502315047854269, −5.50430788075374810390862959900, −4.86910005695865937309811627545, −3.36087545759698411691845364866, −1.45654833213559444421160062712,
2.81530320096720120616464903390, 4.37917765655656044800020305934, 5.77579273700946964593703160217, 6.82409595278538695542902591842, 7.43131457418946568029691413093, 9.123747294892099789992166744539, 9.729361292038115743842935821193, 11.41911455874026717212644702849, 12.26434514625796450903480140639, 13.31112791952232801468976270854
