| L(s) = 1 | + (−1.48 + 1.10i)2-s + (1.05 − 1.37i)3-s + (0.411 − 1.37i)4-s + (0.141 + 2.42i)5-s + (−0.0414 + 3.21i)6-s + (2.45 − 0.980i)7-s + (−0.357 − 0.981i)8-s + (−0.785 − 2.89i)9-s + (−2.89 − 3.44i)10-s + (−1.25 + 1.90i)11-s + (−1.45 − 2.01i)12-s + (3.27 + 2.44i)13-s + (−2.56 + 4.17i)14-s + (3.47 + 2.35i)15-s + (4.01 + 2.64i)16-s + (−0.343 − 1.95i)17-s + ⋯ |
| L(s) = 1 | + (−1.05 + 0.782i)2-s + (0.607 − 0.794i)3-s + (0.205 − 0.688i)4-s + (0.0630 + 1.08i)5-s + (−0.0169 + 1.31i)6-s + (0.928 − 0.370i)7-s + (−0.126 − 0.347i)8-s + (−0.261 − 0.965i)9-s + (−0.913 − 1.08i)10-s + (−0.377 + 0.573i)11-s + (−0.421 − 0.581i)12-s + (0.909 + 0.676i)13-s + (−0.686 + 1.11i)14-s + (0.898 + 0.607i)15-s + (1.00 + 0.660i)16-s + (−0.0834 − 0.472i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.620 - 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.620 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.03090 + 0.498568i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03090 + 0.498568i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.05 + 1.37i)T \) |
| 7 | \( 1 + (-2.45 + 0.980i)T \) |
| good | 2 | \( 1 + (1.48 - 1.10i)T + (0.573 - 1.91i)T^{2} \) |
| 5 | \( 1 + (-0.141 - 2.42i)T + (-4.96 + 0.580i)T^{2} \) |
| 11 | \( 1 + (1.25 - 1.90i)T + (-4.35 - 10.1i)T^{2} \) |
| 13 | \( 1 + (-3.27 - 2.44i)T + (3.72 + 12.4i)T^{2} \) |
| 17 | \( 1 + (0.343 + 1.95i)T + (-15.9 + 5.81i)T^{2} \) |
| 19 | \( 1 + (-2.71 - 0.478i)T + (17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (-0.171 - 0.724i)T + (-20.5 + 10.3i)T^{2} \) |
| 29 | \( 1 + (-2.82 - 1.22i)T + (19.9 + 21.0i)T^{2} \) |
| 31 | \( 1 + (1.54 + 6.53i)T + (-27.7 + 13.9i)T^{2} \) |
| 37 | \( 1 + (6.02 - 5.05i)T + (6.42 - 36.4i)T^{2} \) |
| 41 | \( 1 + (-11.0 - 1.29i)T + (39.8 + 9.45i)T^{2} \) |
| 43 | \( 1 + (-5.83 - 3.83i)T + (17.0 + 39.4i)T^{2} \) |
| 47 | \( 1 + (0.134 + 0.0319i)T + (42.0 + 21.0i)T^{2} \) |
| 53 | \( 1 - 11.4iT - 53T^{2} \) |
| 59 | \( 1 + (-8.03 - 4.03i)T + (35.2 + 47.3i)T^{2} \) |
| 61 | \( 1 + (8.39 - 2.51i)T + (50.9 - 33.5i)T^{2} \) |
| 67 | \( 1 + (-9.66 - 1.12i)T + (65.1 + 15.4i)T^{2} \) |
| 71 | \( 1 + (-1.76 + 4.84i)T + (-54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (2.81 + 0.496i)T + (68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-2.72 - 3.66i)T + (-22.6 + 75.6i)T^{2} \) |
| 83 | \( 1 + (3.87 - 0.453i)T + (80.7 - 19.1i)T^{2} \) |
| 89 | \( 1 + (4.48 + 3.76i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (7.28 + 14.5i)T + (-57.9 + 77.8i)T^{2} \) |
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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
−10.72729757715727952289316793392, −9.705106509748595617541602764343, −8.915598858407054478133477941812, −7.999568762305868412993739163363, −7.39477717112098673539883313470, −6.84587969719393640269190175382, −5.90072665206658923177253580980, −4.09096394051736079863422715898, −2.74961566190919578602959854498, −1.29943850304727599172449028540,
1.08882501257544490093968968841, 2.37517320600795889496422014340, 3.65735296776731849538438900479, 5.05379117512028865302794589112, 5.58535064293233715204354480625, 7.78296434324108371842256454396, 8.519714817881349168848670877619, 8.756904204270711850298319306829, 9.622431563668502908952112799880, 10.76076867057299797090246114310
