| L(s) = 1 | + (−1.06 + 1.26i)2-s + (0.958 + 1.44i)3-s + (−0.124 − 0.708i)4-s + (−2.00 + 0.980i)5-s + (−2.83 − 0.318i)6-s + (1.22 + 0.215i)7-s + (−1.82 − 1.05i)8-s + (−1.16 + 2.76i)9-s + (0.890 − 3.57i)10-s + (3.30 − 1.20i)11-s + (0.902 − 0.859i)12-s + (−0.380 − 0.452i)13-s + (−1.56 + 1.31i)14-s + (−3.34 − 1.95i)15-s + (4.62 − 1.68i)16-s + (−4.46 + 2.58i)17-s + ⋯ |
| L(s) = 1 | + (−0.749 + 0.893i)2-s + (0.553 + 0.832i)3-s + (−0.0624 − 0.354i)4-s + (−0.898 + 0.438i)5-s + (−1.15 − 0.129i)6-s + (0.461 + 0.0814i)7-s + (−0.646 − 0.373i)8-s + (−0.387 + 0.921i)9-s + (0.281 − 1.13i)10-s + (0.996 − 0.362i)11-s + (0.260 − 0.248i)12-s + (−0.105 − 0.125i)13-s + (−0.418 + 0.351i)14-s + (−0.862 − 0.505i)15-s + (1.15 − 0.420i)16-s + (−1.08 + 0.625i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.446i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.177406 + 0.753251i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.177406 + 0.753251i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.958 - 1.44i)T \) |
| 5 | \( 1 + (2.00 - 0.980i)T \) |
| good | 2 | \( 1 + (1.06 - 1.26i)T + (-0.347 - 1.96i)T^{2} \) |
| 7 | \( 1 + (-1.22 - 0.215i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-3.30 + 1.20i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.380 + 0.452i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (4.46 - 2.58i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.41 - 2.45i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.59 + 1.16i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-7.21 - 6.05i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.196 - 1.11i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-4.06 + 2.34i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.43 + 3.71i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.739 - 2.03i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-6.14 - 1.08i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 8.21iT - 53T^{2} \) |
| 59 | \( 1 + (12.6 + 4.62i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (1.38 - 7.85i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (2.82 + 3.37i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (3.43 + 5.95i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (1.23 + 0.715i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.970 + 0.814i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.982 + 1.17i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-6.52 + 11.3i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (4.74 + 13.0i)T + (-74.3 + 62.3i)T^{2} \) |
| 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
−14.28793890702191471782037061601, −12.59646943258281117778801644551, −11.34002248162640230206903447980, −10.47517654789928176425269688795, −8.988399896399612674217003619961, −8.554094770813755762759798103815, −7.48973942440212364939815089621, −6.36204537224308910471324664998, −4.49635997482338435100744386683, −3.23240576328562655685726466373,
1.05849895991160885832202185698, 2.69940778553849779925930321596, 4.45079552171057101179136190477, 6.55458642490521955980300828107, 7.74261907761278248101256720658, 8.831193381191193546989832826491, 9.347841246224489281745563791039, 11.05554712103640621152926849313, 11.68845045124163786919961658440, 12.45590067640292271187649800182
