L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.5 − 0.866i)3-s + (−0.500 − 0.866i)4-s − i·5-s + (−0.866 + 0.499i)6-s + (−1.73 + i)7-s + 3i·8-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (−1.73 − i)11-s − 12-s + 1.99·14-s + (−0.866 − 0.5i)15-s + (0.500 − 0.866i)16-s + (−3.5 − 6.06i)17-s + 0.999i·18-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.288 − 0.499i)3-s + (−0.250 − 0.433i)4-s − 0.447i·5-s + (−0.353 + 0.204i)6-s + (−0.654 + 0.377i)7-s + 1.06i·8-s + (−0.166 − 0.288i)9-s + (−0.158 + 0.273i)10-s + (−0.522 − 0.301i)11-s − 0.288·12-s + 0.534·14-s + (−0.223 − 0.129i)15-s + (0.125 − 0.216i)16-s + (−0.848 − 1.47i)17-s + 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.702 - 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0919839 + 0.220070i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0919839 + 0.220070i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 2 | \( 1 + (0.866 + 0.5i)T + (1 + 1.73i)T^{2} \) |
| 5 | \( 1 + iT - 5T^{2} \) |
| 7 | \( 1 + (1.73 - i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.73 + i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (3.5 + 6.06i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (5.19 - 3i)T + (9.5 - 16.4i)T^{2} \) |
| 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 | \( 1 - 4iT - 31T^{2} \) |
| 37 | \( 1 + (-0.866 - 0.5i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (7.79 + 4.5i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3 - 5.19i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 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 | \( 1 + (-1.73 - i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.19 + 3i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 11iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 14iT - 83T^{2} \) |
| 89 | \( 1 + (12.1 + 7i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.73 - i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.21354403386820869700350371160, −9.347638593869360930629946814412, −8.759812062463938314465638788614, −7.944254099659408466843768717945, −6.70033219465552563070497007774, −5.70069964516609289529625809262, −4.69577444704819325306802975315, −3.00957761668490663918741288685, −1.81578555067958701601834462724, −0.15960876978430283758521300916,
2.53724780255686030195816493210, 3.78992274906571636704038272722, 4.56660881705694473245336617326, 6.34018440805261353762613926376, 6.92289583815323643718417550218, 8.156951532567929898816629174629, 8.642056203754185919737624836704, 9.685933127139917322480740070895, 10.39155141719611864593760789348, 11.04263971560557006081932785799