| L(s) = 1 | + 2.65·3-s + 4.68·7-s + 4.06·9-s + 0.0398·11-s + 1.02·13-s − 4.95·17-s + 8.21·19-s + 12.4·21-s − 4.15·23-s + 2.82·27-s − 0.878·29-s − 5.45·31-s + 0.106·33-s + 9.51·37-s + 2.72·39-s − 5.28·41-s − 8.95·43-s + 9.09·47-s + 14.9·49-s − 13.1·51-s − 1.18·53-s + 21.8·57-s − 9.34·59-s + 2.57·61-s + 19.0·63-s − 8.46·67-s − 11.0·69-s + ⋯ |
| L(s) = 1 | + 1.53·3-s + 1.76·7-s + 1.35·9-s + 0.0120·11-s + 0.284·13-s − 1.20·17-s + 1.88·19-s + 2.71·21-s − 0.866·23-s + 0.544·27-s − 0.163·29-s − 0.980·31-s + 0.0184·33-s + 1.56·37-s + 0.436·39-s − 0.825·41-s − 1.36·43-s + 1.32·47-s + 2.13·49-s − 1.84·51-s − 0.162·53-s + 2.89·57-s − 1.21·59-s + 0.330·61-s + 2.39·63-s − 1.03·67-s − 1.32·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.738685289\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.738685289\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2.65T + 3T^{2} \) |
| 7 | \( 1 - 4.68T + 7T^{2} \) |
| 11 | \( 1 - 0.0398T + 11T^{2} \) |
| 13 | \( 1 - 1.02T + 13T^{2} \) |
| 17 | \( 1 + 4.95T + 17T^{2} \) |
| 19 | \( 1 - 8.21T + 19T^{2} \) |
| 23 | \( 1 + 4.15T + 23T^{2} \) |
| 29 | \( 1 + 0.878T + 29T^{2} \) |
| 31 | \( 1 + 5.45T + 31T^{2} \) |
| 37 | \( 1 - 9.51T + 37T^{2} \) |
| 41 | \( 1 + 5.28T + 41T^{2} \) |
| 43 | \( 1 + 8.95T + 43T^{2} \) |
| 47 | \( 1 - 9.09T + 47T^{2} \) |
| 53 | \( 1 + 1.18T + 53T^{2} \) |
| 59 | \( 1 + 9.34T + 59T^{2} \) |
| 61 | \( 1 - 2.57T + 61T^{2} \) |
| 67 | \( 1 + 8.46T + 67T^{2} \) |
| 71 | \( 1 + 15.0T + 71T^{2} \) |
| 73 | \( 1 + 3.58T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 + 9.65T + 83T^{2} \) |
| 89 | \( 1 - 0.759T + 89T^{2} \) |
| 97 | \( 1 - 6.67T + 97T^{2} \) |
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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
−8.932036562761414859484291283661, −8.447520190900253749220420243898, −7.63686363647662596607332293528, −7.35285149087516641305985300365, −5.91669245336759893381168401162, −4.89542872373031300762202365424, −4.17332433826997082655098985656, −3.21588951568655868795122509135, −2.17602467797532261803727557563, −1.44649272507705907320852538978,
1.44649272507705907320852538978, 2.17602467797532261803727557563, 3.21588951568655868795122509135, 4.17332433826997082655098985656, 4.89542872373031300762202365424, 5.91669245336759893381168401162, 7.35285149087516641305985300365, 7.63686363647662596607332293528, 8.447520190900253749220420243898, 8.932036562761414859484291283661
