| L(s) = 1 | + 2-s − 2.85·3-s + 4-s + 1.11·5-s − 2.85·6-s − 3.01·7-s + 8-s + 5.15·9-s + 1.11·10-s + 4.31·11-s − 2.85·12-s − 6.25·13-s − 3.01·14-s − 3.19·15-s + 16-s + 1.96·17-s + 5.15·18-s − 2.24·19-s + 1.11·20-s + 8.59·21-s + 4.31·22-s + 5.27·23-s − 2.85·24-s − 3.74·25-s − 6.25·26-s − 6.14·27-s − 3.01·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.64·3-s + 0.5·4-s + 0.500·5-s − 1.16·6-s − 1.13·7-s + 0.353·8-s + 1.71·9-s + 0.353·10-s + 1.30·11-s − 0.824·12-s − 1.73·13-s − 0.804·14-s − 0.824·15-s + 0.250·16-s + 0.476·17-s + 1.21·18-s − 0.515·19-s + 0.250·20-s + 1.87·21-s + 0.919·22-s + 1.10·23-s − 0.582·24-s − 0.749·25-s − 1.22·26-s − 1.18·27-s − 0.568·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 2011 | \( 1 - T \) |
| good | 3 | \( 1 + 2.85T + 3T^{2} \) |
| 5 | \( 1 - 1.11T + 5T^{2} \) |
| 7 | \( 1 + 3.01T + 7T^{2} \) |
| 11 | \( 1 - 4.31T + 11T^{2} \) |
| 13 | \( 1 + 6.25T + 13T^{2} \) |
| 17 | \( 1 - 1.96T + 17T^{2} \) |
| 19 | \( 1 + 2.24T + 19T^{2} \) |
| 23 | \( 1 - 5.27T + 23T^{2} \) |
| 29 | \( 1 - 2.55T + 29T^{2} \) |
| 31 | \( 1 + 5.81T + 31T^{2} \) |
| 37 | \( 1 - 9.03T + 37T^{2} \) |
| 41 | \( 1 - 2.89T + 41T^{2} \) |
| 43 | \( 1 - 5.89T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 + 8.44T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 - 8.43T + 61T^{2} \) |
| 67 | \( 1 + 5.46T + 67T^{2} \) |
| 71 | \( 1 - 16.6T + 71T^{2} \) |
| 73 | \( 1 + 9.31T + 73T^{2} \) |
| 79 | \( 1 + 3.16T + 79T^{2} \) |
| 83 | \( 1 + 15.2T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 - 1.09T + 97T^{2} \) |
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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
−7.60646623270908030742601473847, −6.99116728211271909014818368836, −6.25674249143017064964778280232, −6.06543336057243061351888919736, −5.08276186293903882694625662762, −4.56271600076391927327376489990, −3.60162735691978581908852053995, −2.53826812972101835536099165686, −1.29762132933837100490692409302, 0,
1.29762132933837100490692409302, 2.53826812972101835536099165686, 3.60162735691978581908852053995, 4.56271600076391927327376489990, 5.08276186293903882694625662762, 6.06543336057243061351888919736, 6.25674249143017064964778280232, 6.99116728211271909014818368836, 7.60646623270908030742601473847
