| L(s) = 1 | + (1.43 − 1.70i)2-s + (1.69 + 0.364i)3-s + (−0.512 − 2.90i)4-s + (−1.92 + 1.14i)5-s + (3.04 − 2.36i)6-s + (−3.63 − 0.641i)7-s + (−1.83 − 1.06i)8-s + (2.73 + 1.23i)9-s + (−0.805 + 4.91i)10-s + (0.208 − 0.0759i)11-s + (0.193 − 5.11i)12-s + (2.18 + 2.60i)13-s + (−6.29 + 5.28i)14-s + (−3.67 + 1.23i)15-s + (1.11 − 0.405i)16-s + (−3.20 + 1.85i)17-s + ⋯ |
| L(s) = 1 | + (1.01 − 1.20i)2-s + (0.977 + 0.210i)3-s + (−0.256 − 1.45i)4-s + (−0.859 + 0.510i)5-s + (1.24 − 0.965i)6-s + (−1.37 − 0.242i)7-s + (−0.649 − 0.374i)8-s + (0.911 + 0.411i)9-s + (−0.254 + 1.55i)10-s + (0.0629 − 0.0228i)11-s + (0.0557 − 1.47i)12-s + (0.605 + 0.721i)13-s + (−1.68 + 1.41i)14-s + (−0.948 + 0.317i)15-s + (0.278 − 0.101i)16-s + (−0.777 + 0.448i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.356 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.55067 - 1.06854i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.55067 - 1.06854i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.69 - 0.364i)T \) |
| 5 | \( 1 + (1.92 - 1.14i)T \) |
| good | 2 | \( 1 + (-1.43 + 1.70i)T + (-0.347 - 1.96i)T^{2} \) |
| 7 | \( 1 + (3.63 + 0.641i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-0.208 + 0.0759i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-2.18 - 2.60i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (3.20 - 1.85i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.03 + 3.52i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (8.81 - 1.55i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (2.31 + 1.94i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (0.591 + 3.35i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-4.98 + 2.87i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.56 - 1.31i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.988 - 2.71i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-5.41 - 0.954i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 0.995iT - 53T^{2} \) |
| 59 | \( 1 + (-9.07 - 3.30i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (0.770 - 4.37i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (9.78 + 11.6i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (1.34 + 2.33i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.00 - 0.581i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.81 - 3.19i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (2.76 - 3.29i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-1.93 + 3.34i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.14 - 16.8i)T + (-74.3 + 62.3i)T^{2} \) |
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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
−13.21305118718447492565490450738, −12.11427263180469898278149682604, −11.14020410518468266533585290084, −10.16570556304522362934606958429, −9.214932627969611824703114821338, −7.69879413619855893247757459474, −6.35199391211264522311055590886, −4.18823969310901269110334733558, −3.67197014422563482664501375487, −2.46645153659096217605671008409,
3.31112911064811782567801372579, 4.17100270208139757269246871135, 5.82025807024892696938276480308, 6.92316243072446610983283262422, 7.900006424380926222747822339794, 8.782904922339737649517605479309, 10.07131348227309177904216480948, 12.09724909878878019319929505740, 12.83012151250257172944896449273, 13.48106986026078161789802658091
