| L(s) = 1 | + (−2.18 − 1.40i)2-s + (−0.989 + 0.142i)3-s + (1.96 + 4.30i)4-s + (−1.99 − 0.584i)5-s + (2.36 + 1.07i)6-s + (1.49 + 2.18i)7-s + (1.01 − 7.02i)8-s + (0.959 − 0.281i)9-s + (3.52 + 4.06i)10-s + (−0.546 − 0.849i)11-s + (−2.56 − 3.98i)12-s + (2.07 − 1.80i)13-s + (−0.190 − 6.86i)14-s + (2.05 + 0.295i)15-s + (−5.86 + 6.77i)16-s + (−1.29 + 2.84i)17-s + ⋯ |
| L(s) = 1 | + (−1.54 − 0.992i)2-s + (−0.571 + 0.0821i)3-s + (0.983 + 2.15i)4-s + (−0.889 − 0.261i)5-s + (0.963 + 0.440i)6-s + (0.563 + 0.825i)7-s + (0.357 − 2.48i)8-s + (0.319 − 0.0939i)9-s + (1.11 + 1.28i)10-s + (−0.164 − 0.256i)11-s + (−0.739 − 1.15i)12-s + (0.576 − 0.499i)13-s + (−0.0509 − 1.83i)14-s + (0.530 + 0.0762i)15-s + (−1.46 + 1.69i)16-s + (−0.315 + 0.690i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.546 - 0.837i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0203406 + 0.0375565i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0203406 + 0.0375565i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.989 - 0.142i)T \) |
| 7 | \( 1 + (-1.49 - 2.18i)T \) |
| 23 | \( 1 + (-1.55 - 4.53i)T \) |
| good | 2 | \( 1 + (2.18 + 1.40i)T + (0.830 + 1.81i)T^{2} \) |
| 5 | \( 1 + (1.99 + 0.584i)T + (4.20 + 2.70i)T^{2} \) |
| 11 | \( 1 + (0.546 + 0.849i)T + (-4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-2.07 + 1.80i)T + (1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.29 - 2.84i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (2.30 + 5.05i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (0.204 - 0.447i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (6.70 + 0.963i)T + (29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-0.106 - 0.362i)T + (-31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (-1.45 + 4.95i)T + (-34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (8.56 - 1.23i)T + (41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 - 12.1iT - 47T^{2} \) |
| 53 | \( 1 + (0.165 + 0.143i)T + (7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (3.18 - 2.76i)T + (8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (1.30 - 9.06i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-5.08 + 7.90i)T + (-27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (8.80 + 5.65i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (10.4 - 4.79i)T + (47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (5.51 - 4.77i)T + (11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (4.57 - 1.34i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (2.41 + 16.8i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (0.968 + 0.284i)T + (81.6 + 52.4i)T^{2} \) |
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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
−11.12456519781637256102070744837, −10.69292461563322477721583834577, −9.432870487393624897958675415036, −8.692128449327156344957711343642, −8.071592186100304077313538450239, −7.17092565120725184657595620524, −5.73801751722253206039601151398, −4.27832230924244765894689854067, −3.01519025635542172754728547648, −1.55298100803039886981923278341,
0.04850521736439640784400828254, 1.61001174999908947167686833690, 4.04288273088922943949926328340, 5.29514420580609777666520108654, 6.55812658642986448064553919956, 7.12255157578878088559009142977, 7.928513109035270862914592587795, 8.597308610540234101969485753109, 9.765575594802318224612250601046, 10.61044194464318163047742768071
