| L(s) = 1 | + (−0.104 + 0.994i)2-s + (−0.840 − 1.51i)3-s + (−0.978 − 0.207i)4-s + (1.12 − 2.51i)5-s + (1.59 − 0.678i)6-s + (2.40 − 1.09i)7-s + (0.309 − 0.951i)8-s + (−1.58 + 2.54i)9-s + (2.38 + 1.37i)10-s + (−3.22 + 0.777i)11-s + (0.507 + 1.65i)12-s + (0.533 − 0.734i)13-s + (0.835 + 2.51i)14-s + (−4.75 + 0.419i)15-s + (0.913 + 0.406i)16-s + (−0.233 − 2.22i)17-s + ⋯ |
| L(s) = 1 | + (−0.0739 + 0.703i)2-s + (−0.485 − 0.874i)3-s + (−0.489 − 0.103i)4-s + (0.501 − 1.12i)5-s + (0.650 − 0.276i)6-s + (0.910 − 0.413i)7-s + (0.109 − 0.336i)8-s + (−0.528 + 0.848i)9-s + (0.754 + 0.435i)10-s + (−0.972 + 0.234i)11-s + (0.146 + 0.478i)12-s + (0.148 − 0.203i)13-s + (0.223 + 0.670i)14-s + (−1.22 + 0.108i)15-s + (0.228 + 0.101i)16-s + (−0.0566 − 0.538i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0106 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0106 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.732919 - 0.740757i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.732919 - 0.740757i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.104 - 0.994i)T \) |
| 3 | \( 1 + (0.840 + 1.51i)T \) |
| 7 | \( 1 + (-2.40 + 1.09i)T \) |
| 11 | \( 1 + (3.22 - 0.777i)T \) |
| good | 5 | \( 1 + (-1.12 + 2.51i)T + (-3.34 - 3.71i)T^{2} \) |
| 13 | \( 1 + (-0.533 + 0.734i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (0.233 + 2.22i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (1.38 + 6.50i)T + (-17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (4.85 - 2.80i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.35 + 4.17i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (0.217 - 0.0968i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-7.25 + 8.05i)T + (-3.86 - 36.7i)T^{2} \) |
| 41 | \( 1 + (1.32 - 4.07i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 8.42iT - 43T^{2} \) |
| 47 | \( 1 + (2.18 + 10.2i)T + (-42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (-1.15 - 2.60i)T + (-35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (1.31 - 6.17i)T + (-53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (-1.19 + 2.68i)T + (-40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (-2.80 + 4.86i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2.57 + 3.54i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (2.21 - 10.4i)T + (-66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (-4.75 - 0.500i)T + (77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-3.07 + 2.23i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (3.00 - 1.73i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-14.4 - 10.5i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.94721208635281480687762192704, −9.787251201435134452512542876947, −8.739639901972924874471770996960, −7.916570764678767891101007187337, −7.31281916814946766539112604973, −6.04732510137863040211206947895, −5.19509587681590136441881749553, −4.58185918867808917242411135031, −2.16519526283446578447641671963, −0.70443009995605174283311279950,
2.04879963739829580576299140143, 3.25070844523325705656883128901, 4.41175951823356735471983483385, 5.55815138397680489804672964940, 6.29055021067012881222422847175, 7.893154929191723368798313941019, 8.720630593305740319906447760327, 9.984829660788396941382949774435, 10.43600416184908697330440621904, 11.03014155877560198226407830196
