| L(s) = 1 | + 2-s − 3-s + 4-s + 2.65·5-s − 6-s + 0.273·7-s + 8-s − 2·9-s + 2.65·10-s + 4.10·11-s − 12-s − 4.75·13-s + 0.273·14-s − 2.65·15-s + 16-s − 7.57·17-s − 2·18-s − 1.65·19-s + 2.65·20-s − 0.273·21-s + 4.10·22-s − 3.27·23-s − 24-s + 2.02·25-s − 4.75·26-s + 5·27-s + 0.273·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.18·5-s − 0.408·6-s + 0.103·7-s + 0.353·8-s − 0.666·9-s + 0.838·10-s + 1.23·11-s − 0.288·12-s − 1.31·13-s + 0.0732·14-s − 0.684·15-s + 0.250·16-s − 1.83·17-s − 0.471·18-s − 0.378·19-s + 0.592·20-s − 0.0597·21-s + 0.874·22-s − 0.682·23-s − 0.204·24-s + 0.405·25-s − 0.932·26-s + 0.962·27-s + 0.0517·28-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 | \( 1 + T + 3T^{2} \) |
| 5 | \( 1 - 2.65T + 5T^{2} \) |
| 7 | \( 1 - 0.273T + 7T^{2} \) |
| 11 | \( 1 - 4.10T + 11T^{2} \) |
| 13 | \( 1 + 4.75T + 13T^{2} \) |
| 17 | \( 1 + 7.57T + 17T^{2} \) |
| 19 | \( 1 + 1.65T + 19T^{2} \) |
| 23 | \( 1 + 3.27T + 23T^{2} \) |
| 29 | \( 1 + 7.27T + 29T^{2} \) |
| 31 | \( 1 + 8.12T + 31T^{2} \) |
| 37 | \( 1 + 4.10T + 37T^{2} \) |
| 41 | \( 1 - 3.36T + 41T^{2} \) |
| 43 | \( 1 + 5.37T + 43T^{2} \) |
| 47 | \( 1 + 7.35T + 47T^{2} \) |
| 53 | \( 1 + 3.10T + 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 - 5.13T + 61T^{2} \) |
| 67 | \( 1 + 1.26T + 67T^{2} \) |
| 71 | \( 1 - 0.754T + 71T^{2} \) |
| 73 | \( 1 - 7.70T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 + 5.03T + 83T^{2} \) |
| 89 | \( 1 - 3.25T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−8.019716010435238083337271538602, −6.74225695801388547650352843346, −6.68423622058871550570606709741, −5.73272805835513430610896125516, −5.22497831121878040808309733017, −4.43596262246103765783008302457, −3.51994380997967697138954762978, −2.22704356991432371312954094578, −1.87276296044193285885650575282, 0,
1.87276296044193285885650575282, 2.22704356991432371312954094578, 3.51994380997967697138954762978, 4.43596262246103765783008302457, 5.22497831121878040808309733017, 5.73272805835513430610896125516, 6.68423622058871550570606709741, 6.74225695801388547650352843346, 8.019716010435238083337271538602
