| L(s) = 1 | + (1.05 + 0.940i)2-s + (−1.24 + 1.20i)3-s + (0.229 + 1.98i)4-s + (0.128 + 0.646i)5-s + (−2.44 + 0.0961i)6-s + (0.120 − 0.0499i)7-s + (−1.62 + 2.31i)8-s + (0.109 − 2.99i)9-s + (−0.472 + 0.804i)10-s + (−1.51 + 2.27i)11-s + (−2.67 − 2.20i)12-s + (1.27 + 0.252i)13-s + (0.174 + 0.0607i)14-s + (−0.938 − 0.651i)15-s + (−3.89 + 0.911i)16-s + (1.71 − 1.71i)17-s + ⋯ |
| L(s) = 1 | + (0.746 + 0.665i)2-s + (−0.719 + 0.694i)3-s + (0.114 + 0.993i)4-s + (0.0575 + 0.289i)5-s + (−0.999 + 0.0392i)6-s + (0.0455 − 0.0188i)7-s + (−0.575 + 0.817i)8-s + (0.0363 − 0.999i)9-s + (−0.149 + 0.254i)10-s + (−0.457 + 0.685i)11-s + (−0.772 − 0.635i)12-s + (0.352 + 0.0701i)13-s + (0.0465 + 0.0162i)14-s + (−0.242 − 0.168i)15-s + (−0.973 + 0.227i)16-s + (0.416 − 0.416i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.581 - 0.813i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.619533 + 1.20503i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.619533 + 1.20503i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.05 - 0.940i)T \) |
| 3 | \( 1 + (1.24 - 1.20i)T \) |
| good | 5 | \( 1 + (-0.128 - 0.646i)T + (-4.61 + 1.91i)T^{2} \) |
| 7 | \( 1 + (-0.120 + 0.0499i)T + (4.94 - 4.94i)T^{2} \) |
| 11 | \( 1 + (1.51 - 2.27i)T + (-4.20 - 10.1i)T^{2} \) |
| 13 | \( 1 + (-1.27 - 0.252i)T + (12.0 + 4.97i)T^{2} \) |
| 17 | \( 1 + (-1.71 + 1.71i)T - 17iT^{2} \) |
| 19 | \( 1 + (-1.38 - 0.275i)T + (17.5 + 7.27i)T^{2} \) |
| 23 | \( 1 + (-3.71 - 1.53i)T + (16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (-5.35 + 3.57i)T + (11.0 - 26.7i)T^{2} \) |
| 31 | \( 1 + 3.91T + 31T^{2} \) |
| 37 | \( 1 + (-1.85 - 9.30i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (-4.19 + 10.1i)T + (-28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (-4.03 + 6.03i)T + (-16.4 - 39.7i)T^{2} \) |
| 47 | \( 1 + (3.94 + 3.94i)T + 47iT^{2} \) |
| 53 | \( 1 + (7.62 + 5.09i)T + (20.2 + 48.9i)T^{2} \) |
| 59 | \( 1 + (2.03 - 0.404i)T + (54.5 - 22.5i)T^{2} \) |
| 61 | \( 1 + (7.97 - 5.32i)T + (23.3 - 56.3i)T^{2} \) |
| 67 | \( 1 + (-3.96 - 5.93i)T + (-25.6 + 61.8i)T^{2} \) |
| 71 | \( 1 + (-3.68 - 8.90i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-2.46 + 5.95i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (10.3 + 10.3i)T + 79iT^{2} \) |
| 83 | \( 1 + (-2.59 + 13.0i)T + (-76.6 - 31.7i)T^{2} \) |
| 89 | \( 1 + (-0.461 - 1.11i)T + (-62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 + 4.33iT - 97T^{2} \) |
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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
−12.82847370620240584251935792522, −11.99730614533396981799978377635, −11.08371492362513975008623456125, −10.03892685472897733952314121248, −8.845927519509503963896592047398, −7.45490703796799928822239502581, −6.48359383945085940204260196883, −5.36875788675011095565351483670, −4.48249186890234176161647689829, −3.10495138353659965632963983036,
1.18792594347614513277972654657, 2.98874557612312331635062057483, 4.74942032121397349865032938788, 5.67985560793367958164154792358, 6.64349270050534069235773967155, 8.040727808803141575220758263255, 9.423635227636322332490580076518, 10.92684166379288100415113888990, 11.05229053706981361673658345744, 12.53766433202524383087088381057
