| L(s) = 1 | + (−2.18 + 0.385i)2-s + (0.666 + 0.794i)3-s + (2.76 − 1.00i)4-s + (−1.76 − 1.48i)6-s + (−1.75 − 1.01i)7-s + (−1.81 + 1.04i)8-s + (0.334 − 1.89i)9-s + (−0.0424 − 0.0734i)11-s + (2.64 + 1.52i)12-s + (−3.67 + 4.38i)13-s + (4.22 + 1.53i)14-s + (−0.945 + 0.793i)16-s + (−2.49 + 0.439i)17-s + 4.27i·18-s + (−3.21 + 2.94i)19-s + ⋯ |
| L(s) = 1 | + (−1.54 + 0.272i)2-s + (0.384 + 0.458i)3-s + (1.38 − 0.502i)4-s + (−0.720 − 0.604i)6-s + (−0.662 − 0.382i)7-s + (−0.640 + 0.369i)8-s + (0.111 − 0.631i)9-s + (−0.0127 − 0.0221i)11-s + (0.762 + 0.440i)12-s + (−1.01 + 1.21i)13-s + (1.13 + 0.411i)14-s + (−0.236 + 0.198i)16-s + (−0.604 + 0.106i)17-s + 1.00i·18-s + (−0.737 + 0.675i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00117i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00117i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.000106271 + 0.181628i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.000106271 + 0.181628i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 + (3.21 - 2.94i)T \) |
| good | 2 | \( 1 + (2.18 - 0.385i)T + (1.87 - 0.684i)T^{2} \) |
| 3 | \( 1 + (-0.666 - 0.794i)T + (-0.520 + 2.95i)T^{2} \) |
| 7 | \( 1 + (1.75 + 1.01i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.0424 + 0.0734i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (3.67 - 4.38i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (2.49 - 0.439i)T + (15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-0.105 - 0.290i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (0.455 - 2.58i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (4.03 - 6.99i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 5.01iT - 37T^{2} \) |
| 41 | \( 1 + (4.50 - 3.77i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.222 + 0.611i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (6.79 + 1.19i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (4.97 + 13.6i)T + (-40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (1.29 + 7.35i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (12.5 - 4.55i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (8.74 + 1.54i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-13.4 - 4.90i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-6.97 - 8.31i)T + (-12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (0.0887 - 0.0744i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-2.57 - 1.48i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-8.52 - 7.15i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (0.222 - 0.0392i)T + (91.1 - 33.1i)T^{2} \) |
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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
−11.04114400913491065172732911583, −10.08799812832135080727978586836, −9.629581367457136166343647663418, −8.916805471334727581861923499625, −8.128320345095500934835381976451, −6.78946748535328029464068971417, −6.64278590671868298846385991001, −4.66658011135872080505487585115, −3.46120147697809518088059449582, −1.81350607927779991870280745701,
0.16341997551234966670998820019, 2.07169563892624456750267320321, 2.84866845290540393460517324505, 4.82054168806261682970344956915, 6.25796251656101625223455752087, 7.44780612581700266402094319090, 7.82921988101863121214863460714, 8.907262893778811372056558508307, 9.495086547177401209788073054900, 10.47129005264025612806171133686
