L(s) = 1 | + (−1.73 − 0.0404i)3-s + (0.0573 + 0.0573i)5-s + 7-s + (2.99 + 0.139i)9-s + (−1.96 + 1.96i)11-s + (−4.48 − 4.48i)13-s + (−0.0969 − 0.101i)15-s + 4.71i·17-s + (2.60 − 2.60i)19-s + (−1.73 − 0.0404i)21-s + 2.30i·23-s − 4.99i·25-s + (−5.18 − 0.363i)27-s + (3.24 − 3.24i)29-s + 1.26i·31-s + ⋯ |
L(s) = 1 | + (−0.999 − 0.0233i)3-s + (0.0256 + 0.0256i)5-s + 0.377·7-s + (0.998 + 0.0466i)9-s + (−0.592 + 0.592i)11-s + (−1.24 − 1.24i)13-s + (−0.0250 − 0.0262i)15-s + 1.14i·17-s + (0.598 − 0.598i)19-s + (−0.377 − 0.00881i)21-s + 0.479i·23-s − 0.998i·25-s + (−0.997 − 0.0699i)27-s + (0.601 − 0.601i)29-s + 0.227i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.771 + 0.636i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.771 + 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3790240689\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3790240689\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.73 + 0.0404i)T \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + (-0.0573 - 0.0573i)T + 5iT^{2} \) |
| 11 | \( 1 + (1.96 - 1.96i)T - 11iT^{2} \) |
| 13 | \( 1 + (4.48 + 4.48i)T + 13iT^{2} \) |
| 17 | \( 1 - 4.71iT - 17T^{2} \) |
| 19 | \( 1 + (-2.60 + 2.60i)T - 19iT^{2} \) |
| 23 | \( 1 - 2.30iT - 23T^{2} \) |
| 29 | \( 1 + (-3.24 + 3.24i)T - 29iT^{2} \) |
| 31 | \( 1 - 1.26iT - 31T^{2} \) |
| 37 | \( 1 + (2.13 - 2.13i)T - 37iT^{2} \) |
| 41 | \( 1 + 6.69T + 41T^{2} \) |
| 43 | \( 1 + (1.86 + 1.86i)T + 43iT^{2} \) |
| 47 | \( 1 + 5.14T + 47T^{2} \) |
| 53 | \( 1 + (-1.91 - 1.91i)T + 53iT^{2} \) |
| 59 | \( 1 + (-3.22 + 3.22i)T - 59iT^{2} \) |
| 61 | \( 1 + (7.78 + 7.78i)T + 61iT^{2} \) |
| 67 | \( 1 + (8.73 - 8.73i)T - 67iT^{2} \) |
| 71 | \( 1 + 12.9iT - 71T^{2} \) |
| 73 | \( 1 + 8.63iT - 73T^{2} \) |
| 79 | \( 1 + 13.8iT - 79T^{2} \) |
| 83 | \( 1 + (7.95 + 7.95i)T + 83iT^{2} \) |
| 89 | \( 1 + 18.1T + 89T^{2} \) |
| 97 | \( 1 - 4.55T + 97T^{2} \) |
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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
−9.600367635737075006520215497246, −8.257953430706069679637523883454, −7.63561004267651830086821438748, −6.83310480802782390738159804716, −5.85262324857823965897542220750, −5.06570675314225078791692467143, −4.49890913862353386312927191806, −3.04406704282034248442673790211, −1.75763874153660888627130082590, −0.18026340819511904094846113092,
1.42319548588645467436513300650, 2.77905681124501584326403850024, 4.17619426372843713016978058411, 5.07346430734185247691892606616, 5.52688272060879041272792080861, 6.82184110077201183446931882799, 7.20894651603424047544457587908, 8.244722031507127251688455495968, 9.343988189434639242764175530544, 9.928848960193105348646853129897