L(s) = 1 | + (−0.5 − 0.866i)3-s + (−2.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (−1 − 1.73i)11-s − 13-s + (1 + 1.73i)17-s + (2.5 − 4.33i)19-s + (2 + 1.73i)21-s + (−3 + 5.19i)23-s + (2.5 + 4.33i)25-s + 0.999·27-s + 8·29-s + (1.5 + 2.59i)31-s + (−0.999 + 1.73i)33-s + (−4.5 + 7.79i)37-s + ⋯ |
L(s) = 1 | + (−0.288 − 0.499i)3-s + (−0.944 + 0.327i)7-s + (−0.166 + 0.288i)9-s + (−0.301 − 0.522i)11-s − 0.277·13-s + (0.242 + 0.420i)17-s + (0.573 − 0.993i)19-s + (0.436 + 0.377i)21-s + (−0.625 + 1.08i)23-s + (0.5 + 0.866i)25-s + 0.192·27-s + 1.48·29-s + (0.269 + 0.466i)31-s + (−0.174 + 0.301i)33-s + (−0.739 + 1.28i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9816928710\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9816928710\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (2.5 - 0.866i)T \) |
good | 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 + (-1.5 - 2.59i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.5 - 7.79i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + (4 - 6.92i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3 - 5.19i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + (-5.5 - 9.52i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.5 + 4.33i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (6 - 10.3i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 18T + 97T^{2} \) |
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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
−9.736911018352367137765343386626, −8.920591842914791714198456988879, −8.093126505501376770573794505617, −7.16633531331130568745747850705, −6.47453539411374882413216972764, −5.66063025990894928977525669229, −4.84587181434337830202536018489, −3.40446873788660811878111012004, −2.68423668223286424696045265535, −1.13109483239319224487879201996,
0.48043511712562793718704805138, 2.37348506301054889681060003012, 3.44422768050030759813274474217, 4.35983985897812890404926196284, 5.24328481503689659144603232782, 6.24018333968719341857591604218, 6.90340070201399535111045252086, 7.889774898031622798111438929047, 8.763120175094757912631920692655, 9.848589323676962205290741605351