| L(s) = 1 | + (1.07 + 1.68i)2-s + (−1.69 + 3.62i)4-s + (0.755 − 4.28i)5-s + (6.44 + 7.67i)7-s + (−7.93 + 1.04i)8-s + (8.03 − 3.32i)10-s + (16.5 − 2.92i)11-s + (7.70 − 2.80i)13-s + (−6.02 + 19.1i)14-s + (−10.2 − 12.2i)16-s + (−13.2 + 22.9i)17-s + (−10.2 + 5.92i)19-s + (14.2 + 9.98i)20-s + (22.7 + 24.8i)22-s + (−14.3 + 17.1i)23-s + ⋯ |
| L(s) = 1 | + (0.537 + 0.843i)2-s + (−0.422 + 0.906i)4-s + (0.151 − 0.856i)5-s + (0.920 + 1.09i)7-s + (−0.991 + 0.130i)8-s + (0.803 − 0.332i)10-s + (1.50 − 0.265i)11-s + (0.592 − 0.215i)13-s + (−0.430 + 1.36i)14-s + (−0.642 − 0.766i)16-s + (−0.780 + 1.35i)17-s + (−0.540 + 0.311i)19-s + (0.712 + 0.499i)20-s + (1.03 + 1.12i)22-s + (−0.624 + 0.743i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0732 - 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0732 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.61402 + 1.73699i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.61402 + 1.73699i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.07 - 1.68i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-0.755 + 4.28i)T + (-23.4 - 8.55i)T^{2} \) |
| 7 | \( 1 + (-6.44 - 7.67i)T + (-8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (-16.5 + 2.92i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (-7.70 + 2.80i)T + (129. - 108. i)T^{2} \) |
| 17 | \( 1 + (13.2 - 22.9i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (10.2 - 5.92i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (14.3 - 17.1i)T + (-91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (-29.1 - 10.6i)T + (644. + 540. i)T^{2} \) |
| 31 | \( 1 + (12.9 - 15.4i)T + (-166. - 946. i)T^{2} \) |
| 37 | \( 1 + (-20.1 + 34.9i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-34.1 + 12.4i)T + (1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (25.3 - 4.46i)T + (1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (12.9 + 15.4i)T + (-383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 + 46.7T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-36.4 - 6.42i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-48.7 + 40.9i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (25.9 + 71.3i)T + (-3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (17.4 + 10.0i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (50.2 + 87.0i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-29.7 + 81.8i)T + (-4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-25.2 + 69.3i)T + (-5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-37.6 - 65.1i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-12.6 - 71.5i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−11.96444564743805567099086456469, −10.98198127715124706142616273676, −9.177790179038258423894400666826, −8.708218049992145197784031315351, −8.045638838111142724632384597475, −6.46569414761771805883923610030, −5.79047113255680277685878814654, −4.73007057233294203192948305262, −3.73536968810390409394784631961, −1.72160015613128532162715737999,
1.10412702112219654687260092223, 2.54461477700642720369673395413, 4.03488764522298668421739186706, 4.62696794704138014588024443445, 6.35686552470239486123930052564, 6.96003502965827090898192204317, 8.501901680867429289472502747879, 9.597274108075015756781101938323, 10.47605643813274433705285578091, 11.33497909590461859667599505115
