| L(s) = 1 | − 0.760·2-s − 3-s − 1.42·4-s + 0.760·6-s − 2.42·7-s + 2.60·8-s + 9-s + 4.36·11-s + 1.42·12-s + 4.36·13-s + 1.84·14-s + 0.861·16-s − 6.12·17-s − 0.760·18-s − 1.76·19-s + 2.42·21-s − 3.32·22-s + 1.04·23-s − 2.60·24-s − 5·25-s − 3.32·26-s − 27-s + 3.44·28-s − 7.66·29-s − 8.70·31-s − 5.86·32-s − 4.36·33-s + ⋯ |
| L(s) = 1 | − 0.538·2-s − 0.577·3-s − 0.710·4-s + 0.310·6-s − 0.915·7-s + 0.920·8-s + 0.333·9-s + 1.31·11-s + 0.410·12-s + 1.21·13-s + 0.492·14-s + 0.215·16-s − 1.48·17-s − 0.179·18-s − 0.403·19-s + 0.528·21-s − 0.707·22-s + 0.217·23-s − 0.531·24-s − 25-s − 0.651·26-s − 0.192·27-s + 0.650·28-s − 1.42·29-s − 1.56·31-s − 1.03·32-s − 0.759·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 669 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 669 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 223 | \( 1 + T \) |
| good | 2 | \( 1 + 0.760T + 2T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 + 2.42T + 7T^{2} \) |
| 11 | \( 1 - 4.36T + 11T^{2} \) |
| 13 | \( 1 - 4.36T + 13T^{2} \) |
| 17 | \( 1 + 6.12T + 17T^{2} \) |
| 19 | \( 1 + 1.76T + 19T^{2} \) |
| 23 | \( 1 - 1.04T + 23T^{2} \) |
| 29 | \( 1 + 7.66T + 29T^{2} \) |
| 31 | \( 1 + 8.70T + 31T^{2} \) |
| 37 | \( 1 - 6.89T + 37T^{2} \) |
| 41 | \( 1 + 1.46T + 41T^{2} \) |
| 43 | \( 1 - 4.70T + 43T^{2} \) |
| 47 | \( 1 + 3.76T + 47T^{2} \) |
| 53 | \( 1 + 2.60T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 - 15.2T + 61T^{2} \) |
| 67 | \( 1 + 0.363T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 - 1.54T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 + 6.81T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 - 1.63T + 97T^{2} \) |
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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
−9.873380328237099399405595832062, −9.207713314633769441817519756851, −8.689952634725604946617251509821, −7.40891619089674459518733590812, −6.46151717394861042245943996431, −5.75495714697421864835317387662, −4.29508896372631478877612008463, −3.71556637404088055233922968229, −1.60208518210078889155342466779, 0,
1.60208518210078889155342466779, 3.71556637404088055233922968229, 4.29508896372631478877612008463, 5.75495714697421864835317387662, 6.46151717394861042245943996431, 7.40891619089674459518733590812, 8.689952634725604946617251509821, 9.207713314633769441817519756851, 9.873380328237099399405595832062
