| L(s) = 1 | + 3·3-s + (6.39 − 3.69i)5-s + (−3.39 + 5.88i)7-s + 9·9-s + (−5.29 − 3.05i)11-s + (−8.39 − 14.5i)13-s + (19.1 − 11.0i)15-s + 25.1i·17-s + 17.5·19-s + (−10.1 + 17.6i)21-s + (−12.3 + 7.15i)23-s + (14.7 − 25.6i)25-s + 27·27-s + (16.1 + 9.35i)29-s + (−23.3 − 40.5i)31-s + ⋯ |
| L(s) = 1 | + 3-s + (1.27 − 0.738i)5-s + (−0.485 + 0.841i)7-s + 9-s + (−0.481 − 0.278i)11-s + (−0.646 − 1.11i)13-s + (1.27 − 0.738i)15-s + 1.48i·17-s + 0.926·19-s + (−0.485 + 0.841i)21-s + (−0.539 + 0.311i)23-s + (0.591 − 1.02i)25-s + 27-s + (0.558 + 0.322i)29-s + (−0.754 − 1.30i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.16463 - 0.189381i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.16463 - 0.189381i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| good | 5 | \( 1 + (-6.39 + 3.69i)T + (12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (3.39 - 5.88i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (5.29 + 3.05i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (8.39 + 14.5i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 - 25.1iT - 289T^{2} \) |
| 19 | \( 1 - 17.5T + 361T^{2} \) |
| 23 | \( 1 + (12.3 - 7.15i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-16.1 - 9.35i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (23.3 + 40.5i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 49.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + (34.5 - 19.9i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (22.0 - 38.2i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (28.8 + 16.6i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 10.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-14.2 + 8.25i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (10.6 - 18.3i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (43.4 + 75.3i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 30.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 48.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-55.7 + 96.6i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-85.0 - 49.1i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 - 75.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-70.2 + 121. i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−13.00905248234756048706423053751, −12.28628930907100158460170316173, −10.31849739447872205681256167182, −9.676709104197353544369606250021, −8.758332285372489003070324274337, −7.83622947317915995758596225837, −6.09833730928179908222855567453, −5.16054385765402168947639542302, −3.18968364246582292948899385194, −1.87096195090316916790817694638,
2.05922533163694006560593327019, 3.27858143209061309093459445123, 4.97924532786057456527304325791, 6.81030742549333747140828365773, 7.26492404496764337175678030287, 8.991470931175957153758386576243, 9.899729749495428579888247814598, 10.34090924372538328214282119130, 12.00357330465810518060174278512, 13.34843958696432587572967780502
