| L(s) = 1 | − 2-s + 0.801·3-s + 4-s + 3.24·5-s − 0.801·6-s − 2.60·7-s − 8-s − 2.35·9-s − 3.24·10-s − 5.40·11-s + 0.801·12-s + 1.91·13-s + 2.60·14-s + 2.60·15-s + 16-s + 2.85·17-s + 2.35·18-s − 5.13·19-s + 3.24·20-s − 2.08·21-s + 5.40·22-s − 2.02·23-s − 0.801·24-s + 5.54·25-s − 1.91·26-s − 4.29·27-s − 2.60·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.462·3-s + 0.5·4-s + 1.45·5-s − 0.327·6-s − 0.984·7-s − 0.353·8-s − 0.785·9-s − 1.02·10-s − 1.62·11-s + 0.231·12-s + 0.530·13-s + 0.695·14-s + 0.672·15-s + 0.250·16-s + 0.691·17-s + 0.555·18-s − 1.17·19-s + 0.726·20-s − 0.455·21-s + 1.15·22-s − 0.422·23-s − 0.163·24-s + 1.10·25-s − 0.374·26-s − 0.826·27-s − 0.492·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 - 0.801T + 3T^{2} \) |
| 5 | \( 1 - 3.24T + 5T^{2} \) |
| 7 | \( 1 + 2.60T + 7T^{2} \) |
| 11 | \( 1 + 5.40T + 11T^{2} \) |
| 13 | \( 1 - 1.91T + 13T^{2} \) |
| 17 | \( 1 - 2.85T + 17T^{2} \) |
| 19 | \( 1 + 5.13T + 19T^{2} \) |
| 23 | \( 1 + 2.02T + 23T^{2} \) |
| 31 | \( 1 - 1.82T + 31T^{2} \) |
| 37 | \( 1 + 6.76T + 37T^{2} \) |
| 41 | \( 1 + 9.78T + 41T^{2} \) |
| 43 | \( 1 + 2.59T + 43T^{2} \) |
| 47 | \( 1 + 2.24T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 - 8.92T + 61T^{2} \) |
| 67 | \( 1 + 11T + 67T^{2} \) |
| 71 | \( 1 + 6.32T + 71T^{2} \) |
| 73 | \( 1 + 9.32T + 73T^{2} \) |
| 79 | \( 1 - 7.60T + 79T^{2} \) |
| 83 | \( 1 - 7.64T + 83T^{2} \) |
| 89 | \( 1 + 8.10T + 89T^{2} \) |
| 97 | \( 1 - 2.27T + 97T^{2} \) |
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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
−8.823649672611869542540415138271, −8.495557948303382482000956123900, −7.50276723456758748610676962853, −6.43020047396734434453098556679, −5.89905723934871870094660644810, −5.15371270392584344915122101105, −3.40183790634649067847643934554, −2.66060365734896686068647810442, −1.85595880764523973402155825677, 0,
1.85595880764523973402155825677, 2.66060365734896686068647810442, 3.40183790634649067847643934554, 5.15371270392584344915122101105, 5.89905723934871870094660644810, 6.43020047396734434453098556679, 7.50276723456758748610676962853, 8.495557948303382482000956123900, 8.823649672611869542540415138271
