L(s) = 1 | + 0.268·2-s − 1.92·4-s + (−1.28 + 2.21i)5-s + (1.80 − 1.93i)7-s − 1.05·8-s + (−0.343 + 0.594i)10-s + (1.97 − 3.41i)11-s + (−3.15 − 1.74i)13-s + (0.483 − 0.518i)14-s + 3.57·16-s − 0.785·17-s + (3.74 + 6.49i)19-s + (2.46 − 4.27i)20-s + (0.529 − 0.916i)22-s + 7.95·23-s + ⋯ |
L(s) = 1 | + 0.189·2-s − 0.964·4-s + (−0.572 + 0.992i)5-s + (0.681 − 0.731i)7-s − 0.372·8-s + (−0.108 + 0.188i)10-s + (0.594 − 1.03i)11-s + (−0.874 − 0.484i)13-s + (0.129 − 0.138i)14-s + 0.893·16-s − 0.190·17-s + (0.859 + 1.48i)19-s + (0.552 − 0.956i)20-s + (0.112 − 0.195i)22-s + 1.65·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.122i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.29930 + 0.0796270i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.29930 + 0.0796270i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-1.80 + 1.93i)T \) |
| 13 | \( 1 + (3.15 + 1.74i)T \) |
good | 2 | \( 1 - 0.268T + 2T^{2} \) |
| 5 | \( 1 + (1.28 - 2.21i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.97 + 3.41i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + 0.785T + 17T^{2} \) |
| 19 | \( 1 + (-3.74 - 6.49i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 7.95T + 23T^{2} \) |
| 29 | \( 1 + (-1.17 - 2.03i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.27 - 2.21i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 6.75T + 37T^{2} \) |
| 41 | \( 1 + (1.21 + 2.11i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.12 + 1.94i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.658 + 1.14i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.63 - 8.03i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 8.96T + 59T^{2} \) |
| 61 | \( 1 + (4.72 + 8.18i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.676 + 1.17i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.15 + 10.6i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (0.384 + 0.665i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.09 - 5.36i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 1.07T + 83T^{2} \) |
| 89 | \( 1 + 7.66T + 89T^{2} \) |
| 97 | \( 1 + (-1.18 + 2.05i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.37055328989275377723364419861, −9.439063046342151194967087477032, −8.446045064083301191659034831341, −7.70253270924784391156111120538, −6.95342885804616772756266883278, −5.71811266425802233510536606104, −4.80405593509564093649767837692, −3.74069489761454791152321944988, −3.10057409972799192504338787962, −0.945389770868697198052926512062,
0.959570804471549353224845178751, 2.63222610059935418708890490608, 4.28491751512050240978706909365, 4.74321818184931505334761510211, 5.34511778733123592656400128311, 6.87401392633016836081341085530, 7.79000254550402976585093473012, 8.768406017521125739347510129612, 9.180027127256012729964083012243, 9.869233581818648369351231329997