| L(s) = 1 | + (−0.000955 + 1.41i)2-s + (0.368 + 1.69i)3-s + (−1.99 − 0.00270i)4-s + (−0.355 + 0.978i)5-s + (−2.39 + 0.519i)6-s + (−0.272 + 1.54i)7-s + (0.00573 − 2.82i)8-s + (−2.72 + 1.24i)9-s + (−1.38 − 0.504i)10-s + (−1.44 − 3.97i)11-s + (−0.732 − 3.38i)12-s + (4.33 + 5.17i)13-s + (−2.18 − 0.386i)14-s + (−1.78 − 0.242i)15-s + (3.99 + 0.0108i)16-s + (−0.494 − 0.857i)17-s + ⋯ |
| L(s) = 1 | + (−0.000675 + 0.999i)2-s + (0.212 + 0.977i)3-s + (−0.999 − 0.00135i)4-s + (−0.159 + 0.437i)5-s + (−0.977 + 0.211i)6-s + (−0.102 + 0.583i)7-s + (0.00202 − 0.999i)8-s + (−0.909 + 0.415i)9-s + (−0.437 − 0.159i)10-s + (−0.436 − 1.19i)11-s + (−0.211 − 0.977i)12-s + (1.20 + 1.43i)13-s + (−0.583 − 0.103i)14-s + (−0.461 − 0.0625i)15-s + (0.999 + 0.00270i)16-s + (−0.120 − 0.207i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.115i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0595237 + 1.02931i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0595237 + 1.02931i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.000955 - 1.41i)T \) |
| 3 | \( 1 + (-0.368 - 1.69i)T \) |
| good | 5 | \( 1 + (0.355 - 0.978i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (0.272 - 1.54i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (1.44 + 3.97i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-4.33 - 5.17i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.494 + 0.857i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.70 + 1.55i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.17 - 6.68i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-2.84 + 3.39i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.409 + 2.32i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-3.19 + 1.84i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.44 - 2.04i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-3.71 - 10.1i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (0.155 - 0.880i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 5.00iT - 53T^{2} \) |
| 59 | \( 1 + (-3.85 + 10.5i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-5.41 - 0.954i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-0.467 - 0.556i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (2.52 + 4.36i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.01 + 12.1i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6.78 + 5.68i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-4.03 + 4.80i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-4.02 + 6.96i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (10.1 - 3.68i)T + (74.3 - 62.3i)T^{2} \) |
| 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
−13.26530874308828784041407843778, −11.51091679594017993087714746018, −10.87444075778292645342184847315, −9.445567741899670261563400988041, −8.888649845439672709528492528323, −7.966589548347221328036586183071, −6.47142866606665480706202901961, −5.64690547777237781166378557492, −4.34105829038459829692351562154, −3.20647646982859028584862524537,
0.923194228621869642210834884626, 2.53908855114702942184610788278, 3.98061636812951445818856788608, 5.37560937377228306788813229393, 6.86124554210496062367209826694, 8.199232940654775302515632862837, 8.666698160375088202140923127200, 10.25392230224202071950380692225, 10.78549553010478252784725794270, 12.21805305215207695532896619869
