| L(s) = 1 | + (−1.36 − 0.377i)2-s + (−1.69 + 0.344i)3-s + (1.71 + 1.02i)4-s + (0.420 − 1.15i)5-s + (2.44 + 0.170i)6-s + (1.81 + 0.319i)7-s + (−1.94 − 2.05i)8-s + (2.76 − 1.17i)9-s + (−1.01 + 1.41i)10-s + (5.09 − 1.85i)11-s + (−3.26 − 1.15i)12-s + (2.61 − 2.19i)13-s + (−2.35 − 1.12i)14-s + (−0.315 + 2.10i)15-s + (1.88 + 3.53i)16-s + (−4.18 + 2.41i)17-s + ⋯ |
| L(s) = 1 | + (−0.963 − 0.267i)2-s + (−0.979 + 0.199i)3-s + (0.857 + 0.514i)4-s + (0.188 − 0.517i)5-s + (0.997 + 0.0697i)6-s + (0.685 + 0.120i)7-s + (−0.688 − 0.724i)8-s + (0.920 − 0.390i)9-s + (−0.319 + 0.448i)10-s + (1.53 − 0.558i)11-s + (−0.942 − 0.333i)12-s + (0.726 − 0.609i)13-s + (−0.628 − 0.299i)14-s + (−0.0814 + 0.544i)15-s + (0.470 + 0.882i)16-s + (−1.01 + 0.586i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 + 0.497i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.867 + 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.587059 - 0.156442i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.587059 - 0.156442i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.36 + 0.377i)T \) |
| 3 | \( 1 + (1.69 - 0.344i)T \) |
| good | 5 | \( 1 + (-0.420 + 1.15i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-1.81 - 0.319i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-5.09 + 1.85i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-2.61 + 2.19i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (4.18 - 2.41i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.42 + 1.98i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.674 - 3.82i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-1.76 + 2.09i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.190 + 0.0336i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (3.47 + 6.01i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.51 - 3.00i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (2.57 + 7.08i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.343 + 1.94i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 11.2iT - 53T^{2} \) |
| 59 | \( 1 + (3.62 + 1.31i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (2.54 - 14.4i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-1.34 - 1.59i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-4.41 - 7.65i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.62 + 4.53i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.09 - 6.07i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-1.39 - 1.16i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (14.2 + 8.20i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.0876 + 0.0318i)T + (74.3 - 62.3i)T^{2} \) |
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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
−13.28641159097492249759231354936, −12.20625603143692460222248020424, −11.27403038227416782582571516870, −10.70041375230035788127733994320, −9.238819446020300942881382664704, −8.519925121385570325977891492716, −6.87838465512766184273672526428, −5.81531784461529502176104016098, −4.05639147647364610845720078352, −1.33380009138318104809151639388,
1.67874844007809739420730412328, 4.61090582409766556369036180745, 6.51475991433027537307075154864, 6.71995037155137780039434709868, 8.370205179762833452757261769638, 9.572400173557053920295447966151, 10.78970163648225903129634400980, 11.35635391645247055178565801964, 12.34668505293388700353229590958, 14.06356362105056731301389550379
