| L(s) = 1 | + (−0.996 − 0.0871i)2-s + (1.12 − 1.31i)3-s + (0.984 + 0.173i)4-s + (−1.30 + 1.81i)5-s + (−1.23 + 1.21i)6-s + (−3.63 − 2.54i)7-s + (−0.965 − 0.258i)8-s + (−0.483 − 2.96i)9-s + (1.46 − 1.69i)10-s + (1.06 − 2.91i)11-s + (1.33 − 1.10i)12-s + (−0.443 − 5.06i)13-s + (3.39 + 2.85i)14-s + (0.922 + 3.76i)15-s + (0.939 + 0.342i)16-s + (−2.90 + 0.778i)17-s + ⋯ |
| L(s) = 1 | + (−0.704 − 0.0616i)2-s + (0.647 − 0.761i)3-s + (0.492 + 0.0868i)4-s + (−0.585 + 0.810i)5-s + (−0.503 + 0.496i)6-s + (−1.37 − 0.961i)7-s + (−0.341 − 0.0915i)8-s + (−0.161 − 0.986i)9-s + (0.462 − 0.534i)10-s + (0.319 − 0.879i)11-s + (0.385 − 0.318i)12-s + (−0.122 − 1.40i)13-s + (0.908 + 0.762i)14-s + (0.238 + 0.971i)15-s + (0.234 + 0.0855i)16-s + (−0.704 + 0.188i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.505 + 0.863i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.505 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.366714 - 0.639443i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.366714 - 0.639443i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.996 + 0.0871i)T \) |
| 3 | \( 1 + (-1.12 + 1.31i)T \) |
| 5 | \( 1 + (1.30 - 1.81i)T \) |
| good | 7 | \( 1 + (3.63 + 2.54i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (-1.06 + 2.91i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.443 + 5.06i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (2.90 - 0.778i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-4.83 + 2.78i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.26 - 4.66i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (0.945 - 0.793i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.863 - 4.89i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (0.154 + 0.578i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-0.230 + 0.275i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.843 - 1.80i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (-4.62 + 6.60i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (5.61 - 5.61i)T - 53iT^{2} \) |
| 59 | \( 1 + (4.22 - 1.53i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.293 - 1.66i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-8.65 + 0.757i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (-4.54 - 2.62i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.146 + 0.545i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-4.05 - 4.82i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-1.20 + 13.7i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (6.98 + 12.0i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-13.7 + 6.41i)T + (62.3 - 74.3i)T^{2} \) |
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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
−11.44253468434214415254077425057, −10.60289693009784208465753270884, −9.654146502114500644527843332868, −8.672441718431812612391816229037, −7.49513533231460737380069443165, −7.07035958879863498947971698097, −6.06385561115320096041875075984, −3.44669836210506434875014242953, −3.05135428409544367883328721200, −0.64716839361139304019030809017,
2.29481323420064726230465107854, 3.75012004561011786154855289481, 4.94305308826949734031113316698, 6.44241991544440203930145196564, 7.59086865030396193554221553271, 8.816699812213119525976020385312, 9.322260505723927557496466631257, 9.798644524393233141831712196004, 11.27339226486821979853534277622, 12.15457775455885021954783689729
