| L(s) = 1 | + (1.35 + 0.118i)2-s + (0.801 − 1.53i)3-s + (−0.144 − 0.0254i)4-s + (1.16 + 1.91i)5-s + (1.26 − 1.98i)6-s + (0.495 + 0.346i)7-s + (−2.82 − 0.756i)8-s + (−1.71 − 2.46i)9-s + (1.34 + 2.72i)10-s + (−0.461 + 1.26i)11-s + (−0.155 + 0.201i)12-s + (−0.157 − 1.80i)13-s + (0.630 + 0.529i)14-s + (3.86 − 0.253i)15-s + (−3.46 − 1.26i)16-s + (0.596 − 0.159i)17-s + ⋯ |
| L(s) = 1 | + (0.958 + 0.0838i)2-s + (0.462 − 0.886i)3-s + (−0.0722 − 0.0127i)4-s + (0.519 + 0.854i)5-s + (0.518 − 0.811i)6-s + (0.187 + 0.131i)7-s + (−0.998 − 0.267i)8-s + (−0.571 − 0.820i)9-s + (0.426 + 0.862i)10-s + (−0.139 + 0.382i)11-s + (−0.0447 + 0.0581i)12-s + (−0.0437 − 0.499i)13-s + (0.168 + 0.141i)14-s + (0.997 − 0.0655i)15-s + (−0.865 − 0.315i)16-s + (0.144 − 0.0387i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.338i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.941 + 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.73298 - 0.301781i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.73298 - 0.301781i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.801 + 1.53i)T \) |
| 5 | \( 1 + (-1.16 - 1.91i)T \) |
| good | 2 | \( 1 + (-1.35 - 0.118i)T + (1.96 + 0.347i)T^{2} \) |
| 7 | \( 1 + (-0.495 - 0.346i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (0.461 - 1.26i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.157 + 1.80i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (-0.596 + 0.159i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (6.66 - 3.84i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.35 - 6.22i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (-4.15 + 3.48i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.651 + 3.69i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (1.29 + 4.82i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-7.43 + 8.86i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (0.466 + 1.00i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (2.99 - 4.27i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (-0.231 + 0.231i)T - 53iT^{2} \) |
| 59 | \( 1 + (-9.76 + 3.55i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.173 - 0.984i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-1.61 + 0.141i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (8.27 + 4.77i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.09 - 7.80i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (5.86 + 6.98i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (0.973 - 11.1i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (-3.58 - 6.20i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.46 + 4.41i)T + (62.3 - 74.3i)T^{2} \) |
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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
−13.20704878092436722034354280638, −12.61784788403402947500020101849, −11.47471684914479027899577213535, −10.08389667678717828798770495950, −8.911519147754877803986059570659, −7.60522987538613060602469285219, −6.42892603134489583985042420421, −5.53910336288374237663756307545, −3.73081376221076334565353093521, −2.37886990831886313287757452112,
2.77808740351827044287333493702, 4.42221565178092042457384365876, 4.90616321863577152001769558714, 6.30074300467520930705484312092, 8.538167493338130996409644416057, 8.897167373462639188621827235320, 10.19341183779262798494794679453, 11.33619817690539321009276986008, 12.66946670818365189984830300171, 13.31674288657680231744116248412
