L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (0.500 − 0.866i)4-s − 5-s + (0.499 − 0.866i)6-s + (−1 + 1.73i)7-s − 3·8-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)10-s + (1 + 1.73i)11-s + 12-s + (−3.5 + 0.866i)13-s + 1.99·14-s + (−0.5 − 0.866i)15-s + (0.500 + 0.866i)16-s + (3.5 − 6.06i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (0.250 − 0.433i)4-s − 0.447·5-s + (0.204 − 0.353i)6-s + (−0.377 + 0.654i)7-s − 1.06·8-s + (−0.166 + 0.288i)9-s + (0.158 + 0.273i)10-s + (0.301 + 0.522i)11-s + 0.288·12-s + (−0.970 + 0.240i)13-s + 0.534·14-s + (−0.129 − 0.223i)15-s + (0.125 + 0.216i)16-s + (0.848 − 1.47i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.658958 - 0.181103i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.658958 - 0.181103i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (3.5 - 0.866i)T \) |
good | 2 | \( 1 + (0.5 + 0.866i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 + (1 - 1.73i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-3.5 + 6.06i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3 + 5.19i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3 - 5.19i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 + 9T + 53T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1 - 1.73i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 11T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 14T + 83T^{2} \) |
| 89 | \( 1 + (-7 - 12.1i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1 + 1.73i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−15.84408544761433403349579987483, −15.27243809376413943797361286045, −14.02499101548283092459780281457, −12.14357937143744666454386491571, −11.42867064720421448825118723410, −9.791768698484293397611172234114, −9.254398779241532270121720047097, −7.22047791579711791825933304934, −5.20432204657102172725166254858, −2.86393153909535063400258284725,
3.47426894868114625087598578954, 6.24892583756641342435496526807, 7.54477102164470802446718447887, 8.363264865171006164651980684627, 10.08439462454078301203589831316, 11.86794614489910264294294396928, 12.73289175444341950296811916307, 14.25352282411028397136124411118, 15.31436868569418472237139205121, 16.67964485929451701471305897548