L(s) = 1 | − 2.23i·2-s + 2.99·4-s − 5.51·5-s + (4.78 − 17.8i)7-s − 24.5i·8-s + 12.3i·10-s + 34.9i·11-s − 26.7i·13-s + (−40.0 − 10.6i)14-s − 31.0·16-s − 35.9·17-s − 78.0i·19-s − 16.5·20-s + 78.1·22-s − 57.8i·23-s + ⋯ |
L(s) = 1 | − 0.790i·2-s + 0.374·4-s − 0.493·5-s + (0.258 − 0.966i)7-s − 1.08i·8-s + 0.390i·10-s + 0.957i·11-s − 0.570i·13-s + (−0.764 − 0.204i)14-s − 0.485·16-s − 0.512·17-s − 0.942i·19-s − 0.184·20-s + 0.757·22-s − 0.524i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 - 0.258i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.966 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.250660217\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.250660217\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-4.78 + 17.8i)T \) |
good | 2 | \( 1 + 2.23iT - 8T^{2} \) |
| 5 | \( 1 + 5.51T + 125T^{2} \) |
| 11 | \( 1 - 34.9iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 26.7iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 35.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 78.0iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 57.8iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 221. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 145. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 290.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 108.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 348.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 251.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 160. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 734.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 671. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 688.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 577. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 37.4iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 855.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.23e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.50e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.15e3iT - 9.12e5T^{2} \) |
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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
−10.13109123934694878989110134836, −9.257675266897885920747395620319, −7.902316559768356931171583421820, −7.24116514785343840685343471340, −6.42801700581326057162128009165, −4.80259179738718761043640313180, −3.98817664176383833527833306263, −2.87404290187202028053282205992, −1.64362494997300852509525774311, −0.34627417694242575993826724379,
1.78398998373607764263129688243, 2.99856695377092030778291329791, 4.37650154435363944065491430417, 5.68600481480396815693474241052, 6.12712774446732483439406486005, 7.27027119662254839709467259412, 8.171299166852178093382774046270, 8.657607916802960060584161646457, 9.815447492145070413846163930609, 11.10425287641582730358408067902