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
−13.31112791952232801468976270854, −12.26434514625796450903480140639, −11.41911455874026717212644702849, −9.729361292038115743842935821193, −9.123747294892099789992166744539, −7.43131457418946568029691413093, −6.82409595278538695542902591842, −5.77579273700946964593703160217, −4.37917765655656044800020305934, −2.81530320096720120616464903390,
1.45654833213559444421160062712, 3.36087545759698411691845364866, 4.86910005695865937309811627545, 5.50430788075374810390862959900, 7.36525462050641502315047854269, 8.744973192440989079390021927802, 10.04522416590311173316378091647, 10.30280307005648563593139933550, 11.73292902608550261841244714030, 12.69083836112557298240042834324