| L(s) = 1 | + 2-s + 3.18·3-s + 4-s − 1.48·5-s + 3.18·6-s + 8-s + 7.11·9-s − 1.48·10-s + 4.66·11-s + 3.18·12-s + 4.93·13-s − 4.72·15-s + 16-s + 1.79·17-s + 7.11·18-s + 1.06·19-s − 1.48·20-s + 4.66·22-s − 7.90·23-s + 3.18·24-s − 2.79·25-s + 4.93·26-s + 13.0·27-s − 0.551·29-s − 4.72·30-s − 4.11·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.83·3-s + 0.5·4-s − 0.664·5-s + 1.29·6-s + 0.353·8-s + 2.37·9-s − 0.469·10-s + 1.40·11-s + 0.918·12-s + 1.36·13-s − 1.22·15-s + 0.250·16-s + 0.434·17-s + 1.67·18-s + 0.244·19-s − 0.332·20-s + 0.994·22-s − 1.64·23-s + 0.649·24-s − 0.558·25-s + 0.967·26-s + 2.51·27-s − 0.102·29-s − 0.862·30-s − 0.739·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.219884163\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.219884163\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 - 3.18T + 3T^{2} \) |
| 5 | \( 1 + 1.48T + 5T^{2} \) |
| 11 | \( 1 - 4.66T + 11T^{2} \) |
| 13 | \( 1 - 4.93T + 13T^{2} \) |
| 17 | \( 1 - 1.79T + 17T^{2} \) |
| 19 | \( 1 - 1.06T + 19T^{2} \) |
| 23 | \( 1 + 7.90T + 23T^{2} \) |
| 29 | \( 1 + 0.551T + 29T^{2} \) |
| 31 | \( 1 + 4.11T + 31T^{2} \) |
| 37 | \( 1 + 8.36T + 37T^{2} \) |
| 43 | \( 1 + 9.08T + 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 59 | \( 1 - 6.21T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 - 6.05T + 67T^{2} \) |
| 71 | \( 1 - 3.75T + 71T^{2} \) |
| 73 | \( 1 - 5.86T + 73T^{2} \) |
| 79 | \( 1 + 1.50T + 79T^{2} \) |
| 83 | \( 1 + 8.57T + 83T^{2} \) |
| 89 | \( 1 - 3.88T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−8.321569701100787682818168398260, −7.928367464689817436475039781725, −6.99056069847532237426015937352, −6.46490506407866502494651535018, −5.35423870895475935551849602676, −4.04015387013781265883521642375, −3.79111956435046487188883630060, −3.37602969748126326114151060322, −2.09037323240050812357864285836, −1.40027020179370862546417829754,
1.40027020179370862546417829754, 2.09037323240050812357864285836, 3.37602969748126326114151060322, 3.79111956435046487188883630060, 4.04015387013781265883521642375, 5.35423870895475935551849602676, 6.46490506407866502494651535018, 6.99056069847532237426015937352, 7.928367464689817436475039781725, 8.321569701100787682818168398260
