| L(s) = 1 | + (1.19 − 0.752i)2-s + (−1.06 − 1.36i)3-s + (0.868 − 1.80i)4-s + (1.44 − 3.08i)5-s + (−2.30 − 0.836i)6-s + (1.32 − 0.232i)7-s + (−0.313 − 2.81i)8-s + (−0.734 + 2.90i)9-s + (−0.597 − 4.78i)10-s + (1.72 − 0.803i)11-s + (−3.38 + 0.729i)12-s + (0.622 + 7.11i)13-s + (1.40 − 1.27i)14-s + (−5.75 + 1.31i)15-s + (−2.48 − 3.13i)16-s + (−1.10 + 1.91i)17-s + ⋯ |
| L(s) = 1 | + (0.846 − 0.531i)2-s + (−0.614 − 0.788i)3-s + (0.434 − 0.900i)4-s + (0.644 − 1.38i)5-s + (−0.939 − 0.341i)6-s + (0.499 − 0.0879i)7-s + (−0.110 − 0.993i)8-s + (−0.244 + 0.969i)9-s + (−0.189 − 1.51i)10-s + (0.519 − 0.242i)11-s + (−0.977 + 0.210i)12-s + (0.172 + 1.97i)13-s + (0.375 − 0.339i)14-s + (−1.48 + 0.340i)15-s + (−0.622 − 0.782i)16-s + (−0.268 + 0.464i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.668 + 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.668 + 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.858806 - 1.92594i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.858806 - 1.92594i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.19 + 0.752i)T \) |
| 3 | \( 1 + (1.06 + 1.36i)T \) |
| good | 5 | \( 1 + (-1.44 + 3.08i)T + (-3.21 - 3.83i)T^{2} \) |
| 7 | \( 1 + (-1.32 + 0.232i)T + (6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-1.72 + 0.803i)T + (7.07 - 8.42i)T^{2} \) |
| 13 | \( 1 + (-0.622 - 7.11i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (1.10 - 1.91i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.392 - 1.46i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (4.44 + 0.783i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-5.80 - 0.507i)T + (28.5 + 5.03i)T^{2} \) |
| 31 | \( 1 + (-1.12 + 6.35i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-2.50 - 9.34i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-0.557 + 0.664i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (3.37 - 1.57i)T + (27.6 - 32.9i)T^{2} \) |
| 47 | \( 1 + (-0.439 - 2.49i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-2.69 + 2.69i)T - 53iT^{2} \) |
| 59 | \( 1 + (2.21 - 4.74i)T + (-37.9 - 45.1i)T^{2} \) |
| 61 | \( 1 + (-2.36 + 3.37i)T + (-20.8 - 57.3i)T^{2} \) |
| 67 | \( 1 + (0.682 + 7.80i)T + (-65.9 + 11.6i)T^{2} \) |
| 71 | \( 1 + (7.52 + 4.34i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.20 + 3.00i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (12.0 - 10.0i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-14.0 - 1.22i)T + (81.7 + 14.4i)T^{2} \) |
| 89 | \( 1 + (-5.85 + 3.38i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.25 - 2.27i)T + (74.3 + 62.3i)T^{2} \) |
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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
−11.30221782566300481053766885401, −10.06663546790752842119842980871, −9.105018753259602253095246376015, −8.108627062143760955696800076358, −6.54448871811871716694003854610, −6.07952518029084658294047946433, −4.85614926684896016322293106979, −4.27715507704295813620195865014, −2.01296460476090860112962354648, −1.29244561608553012453348998641,
2.66061680979117755841576604291, 3.61328737549754461007846057858, 4.91839230911442239974140446425, 5.78874622016224618024055974337, 6.50376218973171896295672411428, 7.47770334796548413422779241665, 8.678566020250484310430336713037, 10.06913126100780897722595888321, 10.63283667628199072934178776708, 11.43354862151575999979796254469
