| L(s) = 1 | + (0.461 + 1.94i)2-s + (1.72 + 2.45i)3-s + (−3.57 + 1.79i)4-s + (5.25 + 1.04i)5-s + (−3.98 + 4.48i)6-s + (−0.281 + 0.678i)7-s + (−5.14 − 6.12i)8-s + (−3.05 + 8.46i)9-s + (0.390 + 10.7i)10-s + (4.42 + 6.61i)11-s + (−10.5 − 5.68i)12-s + (0.548 + 2.75i)13-s + (−1.45 − 0.233i)14-s + (6.49 + 14.7i)15-s + (9.54 − 12.8i)16-s + (−16.0 − 16.0i)17-s + ⋯ |
| L(s) = 1 | + (0.230 + 0.973i)2-s + (0.574 + 0.818i)3-s + (−0.893 + 0.449i)4-s + (1.05 + 0.209i)5-s + (−0.663 + 0.747i)6-s + (−0.0401 + 0.0969i)7-s + (−0.643 − 0.765i)8-s + (−0.339 + 0.940i)9-s + (0.0390 + 1.07i)10-s + (0.402 + 0.601i)11-s + (−0.880 − 0.473i)12-s + (0.0421 + 0.212i)13-s + (−0.103 − 0.0167i)14-s + (0.432 + 0.980i)15-s + (0.596 − 0.802i)16-s + (−0.942 − 0.942i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 - 0.598i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.654086 + 1.96821i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.654086 + 1.96821i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.461 - 1.94i)T \) |
| 3 | \( 1 + (-1.72 - 2.45i)T \) |
| good | 5 | \( 1 + (-5.25 - 1.04i)T + (23.0 + 9.56i)T^{2} \) |
| 7 | \( 1 + (0.281 - 0.678i)T + (-34.6 - 34.6i)T^{2} \) |
| 11 | \( 1 + (-4.42 - 6.61i)T + (-46.3 + 111. i)T^{2} \) |
| 13 | \( 1 + (-0.548 - 2.75i)T + (-156. + 64.6i)T^{2} \) |
| 17 | \( 1 + (16.0 + 16.0i)T + 289iT^{2} \) |
| 19 | \( 1 + (-14.6 + 2.91i)T + (333. - 138. i)T^{2} \) |
| 23 | \( 1 + (2.52 + 6.08i)T + (-374. + 374. i)T^{2} \) |
| 29 | \( 1 + (-16.9 + 25.4i)T + (-321. - 776. i)T^{2} \) |
| 31 | \( 1 - 25.2iT - 961T^{2} \) |
| 37 | \( 1 + (-36.0 - 7.16i)T + (1.26e3 + 523. i)T^{2} \) |
| 41 | \( 1 + (24.4 + 58.9i)T + (-1.18e3 + 1.18e3i)T^{2} \) |
| 43 | \( 1 + (-8.30 - 12.4i)T + (-707. + 1.70e3i)T^{2} \) |
| 47 | \( 1 + (-44.0 - 44.0i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (29.1 + 43.5i)T + (-1.07e3 + 2.59e3i)T^{2} \) |
| 59 | \( 1 + (-84.6 - 16.8i)T + (3.21e3 + 1.33e3i)T^{2} \) |
| 61 | \( 1 + (38.8 - 58.1i)T + (-1.42e3 - 3.43e3i)T^{2} \) |
| 67 | \( 1 + (-36.3 + 54.3i)T + (-1.71e3 - 4.14e3i)T^{2} \) |
| 71 | \( 1 + (-10.6 - 4.41i)T + (3.56e3 + 3.56e3i)T^{2} \) |
| 73 | \( 1 + (2.21 + 5.35i)T + (-3.76e3 + 3.76e3i)T^{2} \) |
| 79 | \( 1 + (-19.3 - 19.3i)T + 6.24e3iT^{2} \) |
| 83 | \( 1 + (21.7 + 109. i)T + (-6.36e3 + 2.63e3i)T^{2} \) |
| 89 | \( 1 + (-44.5 + 107. i)T + (-5.60e3 - 5.60e3i)T^{2} \) |
| 97 | \( 1 + 154. iT - 9.40e3T^{2} \) |
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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
−13.19845109495817957271066533381, −11.80737680816838952563031590931, −10.30577474881122844243165723724, −9.455882208381338361687622889025, −8.893843142501275071000132303965, −7.53002359958305019172809399980, −6.40543142140870164866742125889, −5.22940591176673141949074949096, −4.22144660213463897097233413841, −2.60848880844525595628220313691,
1.20830443689387459789371736649, 2.42181712214289168176586543447, 3.78767350839277607709823918895, 5.53501097009570583728324104255, 6.48073186559122907886007189054, 8.169869161205432412454944695896, 9.073015711665823740874026593925, 9.864255854735057087666379867442, 11.05799493288324647654443307998, 12.06655093472520003320949368101
