| L(s) = 1 | + (0.237 + 2.71i)2-s + (0.914 + 1.47i)3-s + (−5.34 + 0.942i)4-s + (0.428 − 2.19i)5-s + (−3.77 + 2.83i)6-s + (0.907 + 1.29i)7-s + (−2.41 − 9.01i)8-s + (−1.32 + 2.69i)9-s + (6.05 + 0.642i)10-s + (0.162 + 0.447i)11-s + (−6.27 − 6.99i)12-s + (2.76 + 0.242i)13-s + (−3.30 + 2.77i)14-s + (3.62 − 1.37i)15-s + (13.7 − 4.99i)16-s + (0.898 − 3.35i)17-s + ⋯ |
| L(s) = 1 | + (0.167 + 1.91i)2-s + (0.527 + 0.849i)3-s + (−2.67 + 0.471i)4-s + (0.191 − 0.981i)5-s + (−1.54 + 1.15i)6-s + (0.343 + 0.489i)7-s + (−0.854 − 3.18i)8-s + (−0.442 + 0.896i)9-s + (1.91 + 0.203i)10-s + (0.0491 + 0.134i)11-s + (−1.81 − 2.02i)12-s + (0.767 + 0.0671i)13-s + (−0.882 + 0.740i)14-s + (0.934 − 0.355i)15-s + (3.42 − 1.24i)16-s + (0.217 − 0.813i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.182i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 - 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.111359 + 1.21213i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.111359 + 1.21213i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.914 - 1.47i)T \) |
| 5 | \( 1 + (-0.428 + 2.19i)T \) |
| good | 2 | \( 1 + (-0.237 - 2.71i)T + (-1.96 + 0.347i)T^{2} \) |
| 7 | \( 1 + (-0.907 - 1.29i)T + (-2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (-0.162 - 0.447i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-2.76 - 0.242i)T + (12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (-0.898 + 3.35i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-1.97 - 1.13i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.329 + 0.230i)T + (7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (4.29 + 3.60i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.908 - 5.15i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-1.44 - 0.387i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (4.99 + 5.95i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (1.07 + 0.501i)T + (27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (-9.92 + 6.94i)T + (16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (-0.274 + 0.274i)T - 53iT^{2} \) |
| 59 | \( 1 + (-3.33 - 1.21i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (1.59 - 9.05i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-1.15 + 13.1i)T + (-65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (13.8 - 7.97i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (7.73 - 2.07i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-2.85 + 3.40i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (2.90 - 0.253i)T + (81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (3.50 - 6.06i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3.19 - 6.86i)T + (-62.3 - 74.3i)T^{2} \) |
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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
−13.89307532019215315438774015912, −13.39111626844454269160056136373, −11.99672480291025979615311460890, −9.996830918168163708353859932916, −8.976013865551729870697376772409, −8.536648977117425208172542954314, −7.43322372408133155984832480821, −5.74762061354636814322481724697, −5.06002589911157565716594088321, −3.92458857366921322643389201876,
1.50705909014284960624906650337, 2.91077844680364627373026280123, 3.93188312577412403794763128272, 5.94173040150739671792315932599, 7.66429922183741856041927221912, 8.847450674660794813355655672415, 9.961191788333845803632470772918, 10.96542896122283840085776823842, 11.58733673551454289281534425090, 12.77336584160399374711755562146
