| L(s) = 1 | + 2-s + 1.04·3-s + 4-s − 3.29·5-s + 1.04·6-s + 2.48·7-s + 8-s − 1.90·9-s − 3.29·10-s + 0.839·11-s + 1.04·12-s − 1.54·13-s + 2.48·14-s − 3.44·15-s + 16-s + 3.83·17-s − 1.90·18-s − 5.66·19-s − 3.29·20-s + 2.60·21-s + 0.839·22-s − 6.38·23-s + 1.04·24-s + 5.82·25-s − 1.54·26-s − 5.13·27-s + 2.48·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.604·3-s + 0.5·4-s − 1.47·5-s + 0.427·6-s + 0.939·7-s + 0.353·8-s − 0.634·9-s − 1.04·10-s + 0.253·11-s + 0.302·12-s − 0.428·13-s + 0.664·14-s − 0.889·15-s + 0.250·16-s + 0.930·17-s − 0.448·18-s − 1.29·19-s − 0.735·20-s + 0.567·21-s + 0.178·22-s − 1.33·23-s + 0.213·24-s + 1.16·25-s − 0.303·26-s − 0.988·27-s + 0.469·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 2011 | \( 1 - T \) |
| good | 3 | \( 1 - 1.04T + 3T^{2} \) |
| 5 | \( 1 + 3.29T + 5T^{2} \) |
| 7 | \( 1 - 2.48T + 7T^{2} \) |
| 11 | \( 1 - 0.839T + 11T^{2} \) |
| 13 | \( 1 + 1.54T + 13T^{2} \) |
| 17 | \( 1 - 3.83T + 17T^{2} \) |
| 19 | \( 1 + 5.66T + 19T^{2} \) |
| 23 | \( 1 + 6.38T + 23T^{2} \) |
| 29 | \( 1 + 1.50T + 29T^{2} \) |
| 31 | \( 1 + 4.97T + 31T^{2} \) |
| 37 | \( 1 - 2.78T + 37T^{2} \) |
| 41 | \( 1 - 6.02T + 41T^{2} \) |
| 43 | \( 1 + 4.29T + 43T^{2} \) |
| 47 | \( 1 + 12.6T + 47T^{2} \) |
| 53 | \( 1 - 0.244T + 53T^{2} \) |
| 59 | \( 1 - 4.21T + 59T^{2} \) |
| 61 | \( 1 - 5.10T + 61T^{2} \) |
| 67 | \( 1 + 6.92T + 67T^{2} \) |
| 71 | \( 1 - 7.98T + 71T^{2} \) |
| 73 | \( 1 + 5.94T + 73T^{2} \) |
| 79 | \( 1 - 3.63T + 79T^{2} \) |
| 83 | \( 1 + 5.74T + 83T^{2} \) |
| 89 | \( 1 - 3.75T + 89T^{2} \) |
| 97 | \( 1 + 3.03T + 97T^{2} \) |
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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
−8.032227212475765845024213608252, −7.61086321925384607521205484396, −6.65257143463645369393588546771, −5.71888504503505237878489200613, −4.90043815853823756426423421510, −4.07139291117300319635048315665, −3.64986593599539909726200327338, −2.66679126720237436182587868183, −1.70617222830461016420635719686, 0,
1.70617222830461016420635719686, 2.66679126720237436182587868183, 3.64986593599539909726200327338, 4.07139291117300319635048315665, 4.90043815853823756426423421510, 5.71888504503505237878489200613, 6.65257143463645369393588546771, 7.61086321925384607521205484396, 8.032227212475765845024213608252
