L(s) = 1 | + (2.14 − 0.323i)2-s + (2.58 − 0.797i)4-s + (0.108 − 1.44i)5-s + (−1.12 − 2.39i)7-s + (1.38 − 0.665i)8-s + (−0.235 − 3.14i)10-s + (2.13 − 5.43i)11-s + (4.19 + 5.26i)13-s + (−3.19 − 4.77i)14-s + (−1.72 + 1.17i)16-s + (−0.330 + 0.306i)17-s + (−3.04 + 5.27i)19-s + (−0.874 − 3.83i)20-s + (2.81 − 12.3i)22-s + (1.11 + 1.03i)23-s + ⋯ |
L(s) = 1 | + (1.51 − 0.228i)2-s + (1.29 − 0.398i)4-s + (0.0485 − 0.647i)5-s + (−0.426 − 0.904i)7-s + (0.488 − 0.235i)8-s + (−0.0744 − 0.993i)10-s + (0.642 − 1.63i)11-s + (1.16 + 1.45i)13-s + (−0.853 − 1.27i)14-s + (−0.431 + 0.293i)16-s + (−0.0802 + 0.0744i)17-s + (−0.699 + 1.21i)19-s + (−0.195 − 0.857i)20-s + (0.600 − 2.63i)22-s + (0.231 + 0.215i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.592 + 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.68218 - 1.35595i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.68218 - 1.35595i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (1.12 + 2.39i)T \) |
good | 2 | \( 1 + (-2.14 + 0.323i)T + (1.91 - 0.589i)T^{2} \) |
| 5 | \( 1 + (-0.108 + 1.44i)T + (-4.94 - 0.745i)T^{2} \) |
| 11 | \( 1 + (-2.13 + 5.43i)T + (-8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (-4.19 - 5.26i)T + (-2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (0.330 - 0.306i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (3.04 - 5.27i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.11 - 1.03i)T + (1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-0.284 - 1.24i)T + (-26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (-1.54 - 2.67i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.71 + 0.530i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (4.75 - 2.29i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (6.67 + 3.21i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (3.94 - 0.594i)T + (44.9 - 13.8i)T^{2} \) |
| 53 | \( 1 + (-7.69 + 2.37i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (0.530 + 7.08i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (-0.0365 - 0.0112i)T + (50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (-0.443 - 0.767i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.582 - 2.55i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-5.85 - 0.881i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (-8.35 + 14.4i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.47 + 1.84i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-0.735 - 1.87i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + 5.13T + 97T^{2} \) |
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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
−11.29092904143026621803973135602, −10.49344757382550532887038861423, −9.036191962618974236994785604822, −8.410837768940549278524039794757, −6.62342574740409582039308785927, −6.25849325871486532661351406708, −5.03899867101999059734422831336, −3.88621534105870906036591689743, −3.49265920824331080810570005366, −1.46546320139895328869306480961,
2.42078815808348351736738735029, 3.32939853512109993507054952085, 4.51676212650183767503550851791, 5.48300284984324123637103041481, 6.48799163537186798962998991267, 6.96956820465435386452254995697, 8.453731530482318856358518613364, 9.531694616661940181176835988270, 10.59436670058293600016525049695, 11.55813586004877148612166093822