L(s) = 1 | − 0.456i·2-s + (−1.39 + 2.41i)3-s + 1.79·4-s + (0.395 + 0.228i)5-s + (1.10 + 0.637i)6-s − 1.73i·8-s + (−2.39 − 4.14i)9-s + (0.104 − 0.180i)10-s + (3.39 + 1.96i)11-s + (−2.5 + 4.33i)12-s + (3.5 + 0.866i)13-s + (−1.10 + 0.637i)15-s + 2.79·16-s + 3·17-s + (−1.89 + 1.09i)18-s + (−1.18 + 0.685i)19-s + ⋯ |
L(s) = 1 | − 0.323i·2-s + (−0.805 + 1.39i)3-s + 0.895·4-s + (0.176 + 0.102i)5-s + (0.450 + 0.260i)6-s − 0.612i·8-s + (−0.798 − 1.38i)9-s + (0.0330 − 0.0571i)10-s + (1.02 + 0.591i)11-s + (−0.721 + 1.25i)12-s + (0.970 + 0.240i)13-s + (−0.285 + 0.164i)15-s + 0.697·16-s + 0.727·17-s + (−0.446 + 0.257i)18-s + (−0.272 + 0.157i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.372 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.372 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.29389 + 0.874471i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.29389 + 0.874471i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (-3.5 - 0.866i)T \) |
good | 2 | \( 1 + 0.456iT - 2T^{2} \) |
| 3 | \( 1 + (1.39 - 2.41i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.395 - 0.228i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.39 - 1.96i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 + (1.18 - 0.685i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 1.58T + 23T^{2} \) |
| 29 | \( 1 + (-3.39 - 5.88i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (7.5 - 4.33i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 6.92iT - 37T^{2} \) |
| 41 | \( 1 + (-6.79 + 3.92i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.68 - 8.11i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (8.29 + 4.78i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.08 + 5.33i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 12.3iT - 59T^{2} \) |
| 61 | \( 1 + (-7.37 - 12.7i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.87 - 2.23i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.79 - 2.18i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-3 + 1.73i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3 + 5.19i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 7.02iT - 83T^{2} \) |
| 89 | \( 1 + 16.1iT - 89T^{2} \) |
| 97 | \( 1 + (6.31 + 3.64i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.72434972235414617573360021568, −10.06306393533723756994617105063, −9.452764038797859625091863957229, −8.299408809323139093153817896978, −6.82152337185676039283662164618, −6.23294361841214697552853423746, −5.23009486497388365542472260837, −4.06722926122427706905716311439, −3.33473671278930713498370274384, −1.56922570399140501144258352674,
1.06001833414365627601565033938, 2.08721821007536886643836102810, 3.65128888613961709687098563126, 5.58091600014260064667609010277, 6.02754290170039811582021070218, 6.70849290467544021938648239693, 7.60432479327146753986803404896, 8.258185678958339729471424206629, 9.501379518986480841069363609490, 10.93255889481806058243812868331