| L(s) = 1 | − 2.28·2-s + 3-s + 3.23·4-s − 2.92·5-s − 2.28·6-s − 2.83·8-s + 9-s + 6.69·10-s − 0.608·11-s + 3.23·12-s + 3.41·13-s − 2.92·15-s + 0.00982·16-s + 2.88·17-s − 2.28·18-s − 2.02·19-s − 9.47·20-s + 1.39·22-s − 3.71·23-s − 2.83·24-s + 3.56·25-s − 7.82·26-s + 27-s − 2.68·29-s + 6.69·30-s + 0.845·31-s + 5.64·32-s + ⋯ |
| L(s) = 1 | − 1.61·2-s + 0.577·3-s + 1.61·4-s − 1.30·5-s − 0.934·6-s − 1.00·8-s + 0.333·9-s + 2.11·10-s − 0.183·11-s + 0.934·12-s + 0.948·13-s − 0.755·15-s + 0.00245·16-s + 0.700·17-s − 0.539·18-s − 0.465·19-s − 2.11·20-s + 0.297·22-s − 0.774·23-s − 0.578·24-s + 0.713·25-s − 1.53·26-s + 0.192·27-s − 0.498·29-s + 1.22·30-s + 0.151·31-s + 0.998·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 2.28T + 2T^{2} \) |
| 5 | \( 1 + 2.92T + 5T^{2} \) |
| 11 | \( 1 + 0.608T + 11T^{2} \) |
| 13 | \( 1 - 3.41T + 13T^{2} \) |
| 17 | \( 1 - 2.88T + 17T^{2} \) |
| 19 | \( 1 + 2.02T + 19T^{2} \) |
| 23 | \( 1 + 3.71T + 23T^{2} \) |
| 29 | \( 1 + 2.68T + 29T^{2} \) |
| 31 | \( 1 - 0.845T + 31T^{2} \) |
| 37 | \( 1 - 4.12T + 37T^{2} \) |
| 43 | \( 1 - 0.717T + 43T^{2} \) |
| 47 | \( 1 + 5.20T + 47T^{2} \) |
| 53 | \( 1 - 4.40T + 53T^{2} \) |
| 59 | \( 1 + 8.32T + 59T^{2} \) |
| 61 | \( 1 + 5.34T + 61T^{2} \) |
| 67 | \( 1 - 7.48T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 - 4.75T + 73T^{2} \) |
| 79 | \( 1 - 0.682T + 79T^{2} \) |
| 83 | \( 1 + 1.24T + 83T^{2} \) |
| 89 | \( 1 + 3.56T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−7.87265568739092061644875997532, −7.52057171573824381040523722791, −6.67991887244849747961608402068, −5.90050853354714210080818536592, −4.59898183759069001337128563192, −3.84404841827285601236352234110, −3.11297038968861464038398372020, −2.04397467907434945942065576479, −1.08012962077421461379813935472, 0,
1.08012962077421461379813935472, 2.04397467907434945942065576479, 3.11297038968861464038398372020, 3.84404841827285601236352234110, 4.59898183759069001337128563192, 5.90050853354714210080818536592, 6.67991887244849747961608402068, 7.52057171573824381040523722791, 7.87265568739092061644875997532
