| L(s) = 1 | + (0.766 − 0.642i)2-s + (−1.73 + 0.0664i)3-s + (0.173 − 0.984i)4-s + (−0.294 + 1.66i)5-s + (−1.28 + 1.16i)6-s + (−2.64 − 0.00486i)7-s + (−0.500 − 0.866i)8-s + (2.99 − 0.230i)9-s + (0.847 + 1.46i)10-s + (1.08 + 6.15i)11-s + (−0.235 + 1.71i)12-s + (−0.226 + 1.28i)13-s + (−2.02 + 1.69i)14-s + (0.398 − 2.90i)15-s + (−0.939 − 0.342i)16-s + (−1.03 − 1.79i)17-s + ⋯ |
| L(s) = 1 | + (0.541 − 0.454i)2-s + (−0.999 + 0.0383i)3-s + (0.0868 − 0.492i)4-s + (−0.131 + 0.746i)5-s + (−0.523 + 0.474i)6-s + (−0.999 − 0.00183i)7-s + (−0.176 − 0.306i)8-s + (0.997 − 0.0767i)9-s + (0.268 + 0.464i)10-s + (0.327 + 1.85i)11-s + (−0.0678 + 0.495i)12-s + (−0.0629 + 0.356i)13-s + (−0.542 + 0.453i)14-s + (0.102 − 0.751i)15-s + (−0.234 − 0.0855i)16-s + (−0.251 − 0.435i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.238 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.238 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.695838 + 0.545586i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.695838 + 0.545586i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.766 + 0.642i)T \) |
| 3 | \( 1 + (1.73 - 0.0664i)T \) |
| 7 | \( 1 + (2.64 + 0.00486i)T \) |
| good | 5 | \( 1 + (0.294 - 1.66i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (-1.08 - 6.15i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (0.226 - 1.28i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (1.03 + 1.79i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.71 - 6.44i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.05 - 3.40i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (0.726 + 4.12i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (0.490 - 2.78i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + 9.35T + 37T^{2} \) |
| 41 | \( 1 + (-0.943 + 5.34i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-1.44 + 1.21i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (0.218 + 1.24i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-0.611 + 1.05i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.36 + 0.496i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.48 - 8.39i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (4.26 + 3.58i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-4.83 + 8.37i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 5.04T + 73T^{2} \) |
| 79 | \( 1 + (5.05 - 4.24i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-1.19 - 6.78i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (3.26 - 5.64i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.24 + 6.08i)T + (16.8 - 95.5i)T^{2} \) |
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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
−11.72354740801458485618109740337, −10.59557199133959696099653112091, −10.10756001587208595123355111435, −9.259060647160894568626524918446, −7.16392264947342416208356521322, −6.84561678066371634991116368291, −5.74668527449563143712087424116, −4.55676878735107724555831810272, −3.58465845496239634286078643939, −1.94878378960097914553916681483,
0.55754632351913551497199661833, 3.15337558279908866235379269810, 4.41529233860959529139693151120, 5.41623292688330659228286485144, 6.30450837008000204071058675663, 6.94405793766576702023074748811, 8.478459322898329727040818311463, 9.097610508851900807215431082461, 10.59673283159662228640711857032, 11.20435183995951814556260354857
