| L(s) = 1 | − 2-s − 1.84·3-s + 4-s + 2.20·5-s + 1.84·6-s − 0.412·7-s − 8-s + 0.405·9-s − 2.20·10-s + 0.613·11-s − 1.84·12-s + 3.23·13-s + 0.412·14-s − 4.06·15-s + 16-s − 2.81·17-s − 0.405·18-s − 1.83·19-s + 2.20·20-s + 0.761·21-s − 0.613·22-s + 2.84·23-s + 1.84·24-s − 0.141·25-s − 3.23·26-s + 4.78·27-s − 0.412·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.06·3-s + 0.5·4-s + 0.985·5-s + 0.753·6-s − 0.156·7-s − 0.353·8-s + 0.135·9-s − 0.696·10-s + 0.185·11-s − 0.532·12-s + 0.898·13-s + 0.110·14-s − 1.05·15-s + 0.250·16-s − 0.681·17-s − 0.0956·18-s − 0.421·19-s + 0.492·20-s + 0.166·21-s − 0.130·22-s + 0.593·23-s + 0.376·24-s − 0.0283·25-s − 0.634·26-s + 0.921·27-s − 0.0780·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 2003 | \( 1 + T \) |
| good | 3 | \( 1 + 1.84T + 3T^{2} \) |
| 5 | \( 1 - 2.20T + 5T^{2} \) |
| 7 | \( 1 + 0.412T + 7T^{2} \) |
| 11 | \( 1 - 0.613T + 11T^{2} \) |
| 13 | \( 1 - 3.23T + 13T^{2} \) |
| 17 | \( 1 + 2.81T + 17T^{2} \) |
| 19 | \( 1 + 1.83T + 19T^{2} \) |
| 23 | \( 1 - 2.84T + 23T^{2} \) |
| 29 | \( 1 + 7.99T + 29T^{2} \) |
| 31 | \( 1 - 1.84T + 31T^{2} \) |
| 37 | \( 1 + 5.43T + 37T^{2} \) |
| 41 | \( 1 - 9.53T + 41T^{2} \) |
| 43 | \( 1 - 2.50T + 43T^{2} \) |
| 47 | \( 1 + 3.96T + 47T^{2} \) |
| 53 | \( 1 + 5.16T + 53T^{2} \) |
| 59 | \( 1 + 2.37T + 59T^{2} \) |
| 61 | \( 1 + 0.510T + 61T^{2} \) |
| 67 | \( 1 - 2.29T + 67T^{2} \) |
| 71 | \( 1 + 4.88T + 71T^{2} \) |
| 73 | \( 1 + 14.2T + 73T^{2} \) |
| 79 | \( 1 - 7.76T + 79T^{2} \) |
| 83 | \( 1 + 0.852T + 83T^{2} \) |
| 89 | \( 1 + 0.595T + 89T^{2} \) |
| 97 | \( 1 - 18.9T + 97T^{2} \) |
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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
−8.190659586447072854379774019789, −7.20986736697921930516837427720, −6.41927680947208481183212261407, −6.03407414744055646047202609406, −5.40702400580684156557739157243, −4.43353979465655324588913288288, −3.27129726475285822376056557604, −2.15427456616827587913513530102, −1.26482455512136247382591424744, 0,
1.26482455512136247382591424744, 2.15427456616827587913513530102, 3.27129726475285822376056557604, 4.43353979465655324588913288288, 5.40702400580684156557739157243, 6.03407414744055646047202609406, 6.41927680947208481183212261407, 7.20986736697921930516837427720, 8.190659586447072854379774019789
