| L(s) = 1 | + 0.802·5-s − 1.93i·7-s + 1.56i·11-s + i·13-s − 3.54i·17-s − 4.68·19-s − 8.86·23-s − 4.35·25-s + 8.14·29-s + 10.3i·31-s − 1.55i·35-s + 7.05i·37-s + 4.31i·41-s + 1.21·43-s − 10.6·47-s + ⋯ |
| L(s) = 1 | + 0.359·5-s − 0.732i·7-s + 0.471i·11-s + 0.277i·13-s − 0.860i·17-s − 1.07·19-s − 1.84·23-s − 0.871·25-s + 1.51·29-s + 1.86i·31-s − 0.263i·35-s + 1.15i·37-s + 0.673i·41-s + 0.184·43-s − 1.55·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.492 - 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7847773025\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7847773025\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - iT \) |
| good | 5 | \( 1 - 0.802T + 5T^{2} \) |
| 7 | \( 1 + 1.93iT - 7T^{2} \) |
| 11 | \( 1 - 1.56iT - 11T^{2} \) |
| 17 | \( 1 + 3.54iT - 17T^{2} \) |
| 19 | \( 1 + 4.68T + 19T^{2} \) |
| 23 | \( 1 + 8.86T + 23T^{2} \) |
| 29 | \( 1 - 8.14T + 29T^{2} \) |
| 31 | \( 1 - 10.3iT - 31T^{2} \) |
| 37 | \( 1 - 7.05iT - 37T^{2} \) |
| 41 | \( 1 - 4.31iT - 41T^{2} \) |
| 43 | \( 1 - 1.21T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 + 1.19T + 53T^{2} \) |
| 59 | \( 1 - 8.98iT - 59T^{2} \) |
| 61 | \( 1 - 2.66iT - 61T^{2} \) |
| 67 | \( 1 - 9.46T + 67T^{2} \) |
| 71 | \( 1 + 2.27T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 - 0.0205iT - 79T^{2} \) |
| 83 | \( 1 - 0.956iT - 83T^{2} \) |
| 89 | \( 1 + 4.61iT - 89T^{2} \) |
| 97 | \( 1 - 15.1T + 97T^{2} \) |
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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
−8.620536325875537506264892339423, −8.105249106149322945455036165573, −7.18216322058521672030035303536, −6.59230477287161948898446996356, −5.90947085780694760103827137430, −4.76927256193180284478710177255, −4.34571607059599157238266942647, −3.29238598921055512341283072861, −2.26853043541014820297731894314, −1.30650824646390552801595214707,
0.21856881085861493884704230368, 1.88926224611256883474056907311, 2.43878401873434982472963261315, 3.69134627726917196705340475423, 4.33429898379642005329520539989, 5.49889442280316678657945493894, 6.04862283881234491914275755755, 6.49067222670855456185708277110, 7.82623641969222759473921847876, 8.205074352861282018880252841645
