| L(s) = 1 | + 3-s − 2.41·5-s + 0.521·7-s + 9-s + 1.16·11-s − 13-s − 2.41·15-s − 3.42·17-s + 7.74·19-s + 0.521·21-s + 1.56·23-s + 0.856·25-s + 27-s − 9.30·29-s − 7.35·31-s + 1.16·33-s − 1.26·35-s − 8.33·37-s − 39-s + 6.36·41-s + 11.0·43-s − 2.41·45-s + 11.5·47-s − 6.72·49-s − 3.42·51-s + 10.7·53-s − 2.80·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.08·5-s + 0.197·7-s + 0.333·9-s + 0.349·11-s − 0.277·13-s − 0.624·15-s − 0.830·17-s + 1.77·19-s + 0.113·21-s + 0.326·23-s + 0.171·25-s + 0.192·27-s − 1.72·29-s − 1.32·31-s + 0.202·33-s − 0.213·35-s − 1.37·37-s − 0.160·39-s + 0.994·41-s + 1.69·43-s − 0.360·45-s + 1.68·47-s − 0.961·49-s − 0.479·51-s + 1.47·53-s − 0.378·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 2.41T + 5T^{2} \) |
| 7 | \( 1 - 0.521T + 7T^{2} \) |
| 11 | \( 1 - 1.16T + 11T^{2} \) |
| 17 | \( 1 + 3.42T + 17T^{2} \) |
| 19 | \( 1 - 7.74T + 19T^{2} \) |
| 23 | \( 1 - 1.56T + 23T^{2} \) |
| 29 | \( 1 + 9.30T + 29T^{2} \) |
| 31 | \( 1 + 7.35T + 31T^{2} \) |
| 37 | \( 1 + 8.33T + 37T^{2} \) |
| 41 | \( 1 - 6.36T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + 9.79T + 59T^{2} \) |
| 61 | \( 1 + 0.668T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 - 2.43T + 71T^{2} \) |
| 73 | \( 1 + 5.05T + 73T^{2} \) |
| 79 | \( 1 + 7.13T + 79T^{2} \) |
| 83 | \( 1 - 16.1T + 83T^{2} \) |
| 89 | \( 1 + 8.88T + 89T^{2} \) |
| 97 | \( 1 + 19.6T + 97T^{2} \) |
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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
−7.45290286255844031847327712687, −7.04112961621941503784463117091, −5.84203802151042210018844401002, −5.25975180365676972490099502245, −4.28107087109210466819392205476, −3.83680594692658041429139029236, −3.14314354199265672782446898200, −2.22567428541434939386176305993, −1.23675888421633891542988325151, 0,
1.23675888421633891542988325151, 2.22567428541434939386176305993, 3.14314354199265672782446898200, 3.83680594692658041429139029236, 4.28107087109210466819392205476, 5.25975180365676972490099502245, 5.84203802151042210018844401002, 7.04112961621941503784463117091, 7.45290286255844031847327712687
