| L(s) = 1 | − 1.55·2-s + 1.55·3-s + 0.424·4-s − 1.48·5-s − 2.41·6-s − 1.46·7-s + 2.45·8-s − 0.585·9-s + 2.31·10-s + 1.81·11-s + 0.659·12-s − 4.47·13-s + 2.27·14-s − 2.31·15-s − 4.66·16-s + 2.03·17-s + 0.912·18-s + 2.37·19-s − 0.631·20-s − 2.26·21-s − 2.82·22-s + 9.19·23-s + 3.81·24-s − 2.78·25-s + 6.97·26-s − 5.57·27-s − 0.619·28-s + ⋯ |
| L(s) = 1 | − 1.10·2-s + 0.897·3-s + 0.212·4-s − 0.666·5-s − 0.987·6-s − 0.552·7-s + 0.867·8-s − 0.195·9-s + 0.733·10-s + 0.547·11-s + 0.190·12-s − 1.24·13-s + 0.607·14-s − 0.597·15-s − 1.16·16-s + 0.492·17-s + 0.214·18-s + 0.544·19-s − 0.141·20-s − 0.495·21-s − 0.602·22-s + 1.91·23-s + 0.778·24-s − 0.556·25-s + 1.36·26-s − 1.07·27-s − 0.117·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2671 | \( 1 + T \) |
| good | 2 | \( 1 + 1.55T + 2T^{2} \) |
| 3 | \( 1 - 1.55T + 3T^{2} \) |
| 5 | \( 1 + 1.48T + 5T^{2} \) |
| 7 | \( 1 + 1.46T + 7T^{2} \) |
| 11 | \( 1 - 1.81T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 - 2.03T + 17T^{2} \) |
| 19 | \( 1 - 2.37T + 19T^{2} \) |
| 23 | \( 1 - 9.19T + 23T^{2} \) |
| 29 | \( 1 - 8.73T + 29T^{2} \) |
| 31 | \( 1 - 2.88T + 31T^{2} \) |
| 37 | \( 1 + 9.73T + 37T^{2} \) |
| 41 | \( 1 + 3.23T + 41T^{2} \) |
| 43 | \( 1 - 12.4T + 43T^{2} \) |
| 47 | \( 1 + 9.82T + 47T^{2} \) |
| 53 | \( 1 - 5.77T + 53T^{2} \) |
| 59 | \( 1 + 6.76T + 59T^{2} \) |
| 61 | \( 1 + 3.29T + 61T^{2} \) |
| 67 | \( 1 + 4.88T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 - 6.59T + 73T^{2} \) |
| 79 | \( 1 + 9.26T + 79T^{2} \) |
| 83 | \( 1 + 17.4T + 83T^{2} \) |
| 89 | \( 1 - 4.16T + 89T^{2} \) |
| 97 | \( 1 - 1.27T + 97T^{2} \) |
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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
−8.701532436079185297856700930182, −7.78365391245053466805013203806, −7.36888041978452839133088687575, −6.58738756603903052415274105149, −5.22545135229114557904212949866, −4.40967927396164565573767718047, −3.33230943666587243165940322678, −2.68683545661291413886159400524, −1.29893449219683965281810487492, 0,
1.29893449219683965281810487492, 2.68683545661291413886159400524, 3.33230943666587243165940322678, 4.40967927396164565573767718047, 5.22545135229114557904212949866, 6.58738756603903052415274105149, 7.36888041978452839133088687575, 7.78365391245053466805013203806, 8.701532436079185297856700930182
