| L(s) = 1 | + 1.61·3-s + 11.7·7-s − 24.3·9-s + 32.3·11-s + 33.7·13-s − 95.6·17-s + 75.8·19-s + 19.0·21-s − 40.2·23-s − 83.1·27-s − 269.·29-s − 115.·31-s + 52.3·33-s + 1.68·37-s + 54.6·39-s − 271.·41-s + 119.·43-s − 38.3·47-s − 204.·49-s − 154.·51-s + 698.·53-s + 122.·57-s + 642.·59-s − 176.·61-s − 286.·63-s − 265.·67-s − 65.1·69-s + ⋯ |
| L(s) = 1 | + 0.311·3-s + 0.635·7-s − 0.903·9-s + 0.886·11-s + 0.720·13-s − 1.36·17-s + 0.915·19-s + 0.197·21-s − 0.365·23-s − 0.592·27-s − 1.72·29-s − 0.667·31-s + 0.276·33-s + 0.00747·37-s + 0.224·39-s − 1.03·41-s + 0.425·43-s − 0.118·47-s − 0.596·49-s − 0.425·51-s + 1.81·53-s + 0.285·57-s + 1.41·59-s − 0.371·61-s − 0.573·63-s − 0.483·67-s − 0.113·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2000 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 1.61T + 27T^{2} \) |
| 7 | \( 1 - 11.7T + 343T^{2} \) |
| 11 | \( 1 - 32.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 33.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 95.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 75.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 40.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 269.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 115.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 1.68T + 5.06e4T^{2} \) |
| 41 | \( 1 + 271.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 119.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 38.3T + 1.03e5T^{2} \) |
| 53 | \( 1 - 698.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 642.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 176.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 265.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 155.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 41.7T + 3.89e5T^{2} \) |
| 79 | \( 1 + 983.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.25e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.10e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.34e3T + 9.12e5T^{2} \) |
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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
−8.695713106629250325829063684829, −7.67473322869089856026390159269, −6.90612681694090464580851934372, −5.95439747516803420255170255750, −5.28650157619050388789863271389, −4.14845378079605867000121452743, −3.48035899028942776565934078903, −2.29525312844007969392796393411, −1.40465554920942145747495468588, 0,
1.40465554920942145747495468588, 2.29525312844007969392796393411, 3.48035899028942776565934078903, 4.14845378079605867000121452743, 5.28650157619050388789863271389, 5.95439747516803420255170255750, 6.90612681694090464580851934372, 7.67473322869089856026390159269, 8.695713106629250325829063684829
