L(s) = 1 | + (−3.03 − 1.75i)2-s + (−2.90 + 0.729i)3-s + (4.14 + 7.18i)4-s + (1.07 + 0.620i)5-s + (10.1 + 2.88i)6-s − 15.0i·8-s + (7.93 − 4.24i)9-s + (−2.17 − 3.77i)10-s + (6.07 − 3.50i)11-s + (−17.3 − 17.8i)12-s − 11.6·13-s + (−3.58 − 1.02i)15-s + (−9.79 + 16.9i)16-s + (−3.92 + 2.26i)17-s + (−31.5 − 1.01i)18-s + (−8.11 + 14.0i)19-s + ⋯ |
L(s) = 1 | + (−1.51 − 0.876i)2-s + (−0.969 + 0.243i)3-s + (1.03 + 1.79i)4-s + (0.215 + 0.124i)5-s + (1.68 + 0.480i)6-s − 1.88i·8-s + (0.881 − 0.471i)9-s + (−0.217 − 0.377i)10-s + (0.552 − 0.318i)11-s + (−1.44 − 1.48i)12-s − 0.895·13-s + (−0.238 − 0.0681i)15-s + (−0.611 + 1.05i)16-s + (−0.230 + 0.133i)17-s + (−1.75 − 0.0562i)18-s + (−0.427 + 0.739i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.118i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.992 + 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0108039 - 0.181372i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0108039 - 0.181372i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.90 - 0.729i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (3.03 + 1.75i)T + (2 + 3.46i)T^{2} \) |
| 5 | \( 1 + (-1.07 - 0.620i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (-6.07 + 3.50i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 11.6T + 169T^{2} \) |
| 17 | \( 1 + (3.92 - 2.26i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (8.11 - 14.0i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (22.1 + 12.7i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 9.49iT - 841T^{2} \) |
| 31 | \( 1 + (14.3 + 24.8i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-16.5 + 28.6i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 67.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 24.1T + 1.84e3T^{2} \) |
| 47 | \( 1 + (28.5 + 16.5i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-13.1 + 7.59i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (80.0 - 46.1i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-28.7 + 49.8i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (7.58 + 13.1i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 70.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-38.3 - 66.5i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (63.6 - 110. i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 74.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (110. + 63.6i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + 23.1T + 9.40e3T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.97227887444634510089314895035, −11.13255922945683134524146190570, −10.19936207345448722223497215264, −9.645863308872989437228445124767, −8.414751464236782130905631354959, −7.19860963579272158598811083667, −5.94758707377707452018523061498, −4.03126029465694667871353361712, −2.05152047959341829094155208316, −0.21585163968997657416257369883,
1.58960190515119101965955492115, 4.89858060578249987895281559708, 6.16652359603017392553914301241, 6.99789138261953900312552516629, 7.907712196920756402052019423473, 9.291837665412079266723409281621, 9.958097454132198131475488591339, 11.02627902024638129315538894176, 11.98002491310751919571054576651, 13.24832187750562409689514254443