| L(s) = 1 | + (−0.515 + 1.93i)2-s + (0.149 + 2.99i)3-s + (−3.46 − 1.99i)4-s + (−6.20 − 1.23i)5-s + (−5.86 − 1.25i)6-s + (1.70 − 4.11i)7-s + (5.64 − 5.67i)8-s + (−8.95 + 0.895i)9-s + (5.58 − 11.3i)10-s + (4.91 + 7.36i)11-s + (5.45 − 10.6i)12-s + (−4.36 − 21.9i)13-s + (7.07 + 5.41i)14-s + (2.77 − 18.7i)15-s + (8.04 + 13.8i)16-s + (0.122 + 0.122i)17-s + ⋯ |
| L(s) = 1 | + (−0.257 + 0.966i)2-s + (0.0498 + 0.998i)3-s + (−0.866 − 0.498i)4-s + (−1.24 − 0.246i)5-s + (−0.977 − 0.209i)6-s + (0.243 − 0.587i)7-s + (0.705 − 0.708i)8-s + (−0.995 + 0.0994i)9-s + (0.558 − 1.13i)10-s + (0.447 + 0.669i)11-s + (0.454 − 0.890i)12-s + (−0.335 − 1.68i)13-s + (0.505 + 0.386i)14-s + (0.184 − 1.25i)15-s + (0.503 + 0.864i)16-s + (0.00723 + 0.00723i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.519 + 0.854i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.519 + 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.233415 - 0.131308i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.233415 - 0.131308i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.515 - 1.93i)T \) |
| 3 | \( 1 + (-0.149 - 2.99i)T \) |
| good | 5 | \( 1 + (6.20 + 1.23i)T + (23.0 + 9.56i)T^{2} \) |
| 7 | \( 1 + (-1.70 + 4.11i)T + (-34.6 - 34.6i)T^{2} \) |
| 11 | \( 1 + (-4.91 - 7.36i)T + (-46.3 + 111. i)T^{2} \) |
| 13 | \( 1 + (4.36 + 21.9i)T + (-156. + 64.6i)T^{2} \) |
| 17 | \( 1 + (-0.122 - 0.122i)T + 289iT^{2} \) |
| 19 | \( 1 + (14.6 - 2.91i)T + (333. - 138. i)T^{2} \) |
| 23 | \( 1 + (1.69 + 4.09i)T + (-374. + 374. i)T^{2} \) |
| 29 | \( 1 + (25.7 - 38.5i)T + (-321. - 776. i)T^{2} \) |
| 31 | \( 1 + 53.5iT - 961T^{2} \) |
| 37 | \( 1 + (-10.3 - 2.05i)T + (1.26e3 + 523. i)T^{2} \) |
| 41 | \( 1 + (23.4 + 56.6i)T + (-1.18e3 + 1.18e3i)T^{2} \) |
| 43 | \( 1 + (19.0 + 28.4i)T + (-707. + 1.70e3i)T^{2} \) |
| 47 | \( 1 + (28.0 + 28.0i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (-6.02 - 9.01i)T + (-1.07e3 + 2.59e3i)T^{2} \) |
| 59 | \( 1 + (50.6 + 10.0i)T + (3.21e3 + 1.33e3i)T^{2} \) |
| 61 | \( 1 + (42.8 - 64.0i)T + (-1.42e3 - 3.43e3i)T^{2} \) |
| 67 | \( 1 + (49.5 - 74.2i)T + (-1.71e3 - 4.14e3i)T^{2} \) |
| 71 | \( 1 + (75.5 + 31.2i)T + (3.56e3 + 3.56e3i)T^{2} \) |
| 73 | \( 1 + (-4.66 - 11.2i)T + (-3.76e3 + 3.76e3i)T^{2} \) |
| 79 | \( 1 + (-25.4 - 25.4i)T + 6.24e3iT^{2} \) |
| 83 | \( 1 + (-15.8 - 79.6i)T + (-6.36e3 + 2.63e3i)T^{2} \) |
| 89 | \( 1 + (1.58 - 3.83i)T + (-5.60e3 - 5.60e3i)T^{2} \) |
| 97 | \( 1 + 1.39iT - 9.40e3T^{2} \) |
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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
−12.14427057899249094894776194445, −10.82418967938986197221707035812, −10.13779616692341146870558407650, −8.946508182709851216933408534145, −8.028370594717955944994483965538, −7.29180142621189919575448662025, −5.63815061861552350053197385388, −4.52582388393801313956805422517, −3.72485769349856854515307895131, −0.16993230315243015009017799902,
1.78345290958160896792451423556, 3.25500728732311495733745006059, 4.55765023232878453352445905045, 6.41215881394893161847626359310, 7.63085116407756827940558600634, 8.486101825867783243589923350004, 9.285187507393331503186661171075, 11.01344833144472102922943451948, 11.71490206550202758076602208042, 12.00752939328556415872548698071
