| L(s) = 1 | + (7.14 + 4.12i)2-s + (−8.42 + 13.1i)3-s + (18.0 + 31.1i)4-s + (−3.57 + 6.18i)5-s + (−114. + 58.8i)6-s + (122.5 + 42.4i)7-s + 32.9i·8-s + (−100. − 221. i)9-s + (−51 + 29.4i)10-s + (389. − 224. i)11-s + (−560. − 26.7i)12-s + 523. i·13-s + (699. + 808. i)14-s + (−50.9 − 98.9i)15-s + (439. − 762. i)16-s + (−592. − 1.02e3i)17-s + ⋯ |
| L(s) = 1 | + (1.26 + 0.728i)2-s + (−0.540 + 0.841i)3-s + (0.562 + 0.974i)4-s + (−0.0638 + 0.110i)5-s + (−1.29 + 0.667i)6-s + (0.944 + 0.327i)7-s + 0.182i·8-s + (−0.415 − 0.909i)9-s + (−0.161 + 0.0931i)10-s + (0.969 − 0.559i)11-s + (−1.12 − 0.0536i)12-s + 0.858i·13-s + (0.954 + 1.10i)14-s + (−0.0585 − 0.113i)15-s + (0.429 − 0.744i)16-s + (−0.497 − 0.861i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0156 - 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0156 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.60848 + 1.63386i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.60848 + 1.63386i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (8.42 - 13.1i)T \) |
| 7 | \( 1 + (-122.5 - 42.4i)T \) |
| good | 2 | \( 1 + (-7.14 - 4.12i)T + (16 + 27.7i)T^{2} \) |
| 5 | \( 1 + (3.57 - 6.18i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (-389. + 224. i)T + (8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 - 523. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + (592. + 1.02e3i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (471 + 271. i)T + (1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (3.29e3 + 1.90e3i)T + (3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 - 2.62e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + (7.51e3 - 4.33e3i)T + (1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-3.05e3 + 5.28e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + 199.T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.74e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.30e4 + 2.26e4i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (1.19e4 - 6.87e3i)T + (2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (1.45e4 + 2.51e4i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (3.85e3 + 2.22e3i)T + (4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (691 + 1.19e3i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 - 8.32e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (-2.81e4 + 1.62e4i)T + (1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (1.05e4 - 1.82e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 - 3.66e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (3.07e4 - 5.32e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 - 1.20e5iT - 8.58e9T^{2} \) |
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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
−16.85934139710594784623513425201, −15.98047856388217276189476430429, −14.73155119348648437117860751430, −14.09226847401836089544882179176, −12.16768138410973053336396405033, −11.13006909321451093930083160594, −9.052448337966775250062756264611, −6.71378205406842123503667224457, −5.26795739969227963958493665748, −4.00282045043706396501540201181,
1.78117345881310520900385770984, 4.34624893800283001571296392263, 5.97008250106093372139424855013, 7.961824502364955196678881411007, 10.72940445672658867391721045390, 11.80028000911538675374951070508, 12.72033307333396026773846327771, 13.88407775893363777938964606385, 14.94568025812621349852359941696, 17.07813385698061599516745002103
