| L(s) = 1 | + (0.245 + 1.39i)2-s + (2.87 − 0.847i)3-s + (−1.87 + 0.684i)4-s + (−3.77 − 0.665i)5-s + (1.88 + 3.80i)6-s + (−6.92 + 1.05i)7-s + (−1.41 − 2.44i)8-s + (7.56 − 4.87i)9-s − 5.42i·10-s + (2.47 + 14.0i)11-s + (−4.82 + 3.56i)12-s + (−8.09 + 9.65i)13-s + (−3.16 − 9.37i)14-s + (−11.4 + 1.28i)15-s + (3.06 − 2.57i)16-s + 17.1i·17-s + ⋯ |
| L(s) = 1 | + (0.122 + 0.696i)2-s + (0.959 − 0.282i)3-s + (−0.469 + 0.171i)4-s + (−0.754 − 0.133i)5-s + (0.314 + 0.633i)6-s + (−0.988 + 0.150i)7-s + (−0.176 − 0.306i)8-s + (0.840 − 0.541i)9-s − 0.542i·10-s + (0.224 + 1.27i)11-s + (−0.402 + 0.296i)12-s + (−0.622 + 0.742i)13-s + (−0.226 − 0.669i)14-s + (−0.761 + 0.0855i)15-s + (0.191 − 0.160i)16-s + 1.01i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.898 - 0.439i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.898 - 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.256616 + 1.10897i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.256616 + 1.10897i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.245 - 1.39i)T \) |
| 3 | \( 1 + (-2.87 + 0.847i)T \) |
| 7 | \( 1 + (6.92 - 1.05i)T \) |
| good | 5 | \( 1 + (3.77 + 0.665i)T + (23.4 + 8.55i)T^{2} \) |
| 11 | \( 1 + (-2.47 - 14.0i)T + (-113. + 41.3i)T^{2} \) |
| 13 | \( 1 + (8.09 - 9.65i)T + (-29.3 - 166. i)T^{2} \) |
| 17 | \( 1 - 17.1iT - 289T^{2} \) |
| 19 | \( 1 - 25.5iT - 361T^{2} \) |
| 23 | \( 1 + (21.2 + 17.8i)T + (91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (28.4 - 23.8i)T + (146. - 828. i)T^{2} \) |
| 31 | \( 1 + (2.63 + 7.22i)T + (-736. + 617. i)T^{2} \) |
| 37 | \( 1 + (-18.2 - 31.5i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-35.6 + 42.5i)T + (-291. - 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-15.3 - 5.58i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-7.60 + 20.8i)T + (-1.69e3 - 1.41e3i)T^{2} \) |
| 53 | \( 1 + (17.5 + 30.3i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-32.4 + 38.6i)T + (-604. - 3.42e3i)T^{2} \) |
| 61 | \( 1 + (21.0 - 57.8i)T + (-2.85e3 - 2.39e3i)T^{2} \) |
| 67 | \( 1 + (18.0 - 102. i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (-44.9 + 77.8i)T + (-2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (61.0 + 35.2i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-17.4 - 98.7i)T + (-5.86e3 + 2.13e3i)T^{2} \) |
| 83 | \( 1 + (44.3 + 52.7i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + 13.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-30.1 + 82.9i)T + (-7.20e3 - 6.04e3i)T^{2} \) |
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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
−12.10228350309285360947615333686, −10.18172775376557138854447227963, −9.566428854568342314241289580131, −8.607317832095408438776087936882, −7.70810631350624756337155898204, −7.02084999626043888570699265968, −6.04775547742208996669774616835, −4.29665930361222093515927205010, −3.72773323354295121165444371252, −2.05596029849395402604017071499,
0.41783000354299449761143120529, 2.66127728778500002936439176097, 3.37361739393817439670364051301, 4.30301641161422187343398351058, 5.74936116442566315477974811496, 7.27433044626342961788298619866, 8.021701244796720006352929925644, 9.265408176532279636925921721923, 9.623817080863704035972712731329, 10.83742595824260612422401208702
