| L(s) = 1 | + 2.56·2-s − 3-s + 4.57·4-s − 2.56·6-s − 4.44·7-s + 6.59·8-s + 9-s − 5.41·11-s − 4.57·12-s + 5.68·13-s − 11.4·14-s + 7.76·16-s + 0.388·17-s + 2.56·18-s − 5.51·19-s + 4.44·21-s − 13.8·22-s − 3.44·23-s − 6.59·24-s + 14.5·26-s − 27-s − 20.3·28-s + 1.30·29-s + 3.60·31-s + 6.72·32-s + 5.41·33-s + 0.996·34-s + ⋯ |
| L(s) = 1 | + 1.81·2-s − 0.577·3-s + 2.28·4-s − 1.04·6-s − 1.68·7-s + 2.33·8-s + 0.333·9-s − 1.63·11-s − 1.32·12-s + 1.57·13-s − 3.04·14-s + 1.94·16-s + 0.0942·17-s + 0.604·18-s − 1.26·19-s + 0.970·21-s − 2.95·22-s − 0.717·23-s − 1.34·24-s + 2.85·26-s − 0.192·27-s − 3.84·28-s + 0.242·29-s + 0.647·31-s + 1.18·32-s + 0.942·33-s + 0.170·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 - 2.56T + 2T^{2} \) |
| 7 | \( 1 + 4.44T + 7T^{2} \) |
| 11 | \( 1 + 5.41T + 11T^{2} \) |
| 13 | \( 1 - 5.68T + 13T^{2} \) |
| 17 | \( 1 - 0.388T + 17T^{2} \) |
| 19 | \( 1 + 5.51T + 19T^{2} \) |
| 23 | \( 1 + 3.44T + 23T^{2} \) |
| 29 | \( 1 - 1.30T + 29T^{2} \) |
| 31 | \( 1 - 3.60T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 + 7.19T + 41T^{2} \) |
| 43 | \( 1 + 4.48T + 43T^{2} \) |
| 53 | \( 1 - 2.35T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 + 1.24T + 67T^{2} \) |
| 71 | \( 1 - 1.25T + 71T^{2} \) |
| 73 | \( 1 + 4.03T + 73T^{2} \) |
| 79 | \( 1 - 1.18T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 - 5.78T + 89T^{2} \) |
| 97 | \( 1 + 5.11T + 97T^{2} \) |
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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
−7.946830967099318201351386824476, −6.91624318577252188254848753960, −6.35356759120967320685393239642, −5.95900144614382541038203248917, −5.21736125240025380919448953295, −4.36080756102071648217345175073, −3.49635718241382694524558806585, −3.01108706317753123568515569384, −1.91156799650231233859437154185, 0,
1.91156799650231233859437154185, 3.01108706317753123568515569384, 3.49635718241382694524558806585, 4.36080756102071648217345175073, 5.21736125240025380919448953295, 5.95900144614382541038203248917, 6.35356759120967320685393239642, 6.91624318577252188254848753960, 7.946830967099318201351386824476
