| L(s) = 1 | + (−1.23 + 3.80i)2-s + (7.28 + 5.29i)3-s + (−12.9 − 9.40i)4-s + (−13.4 − 54.2i)5-s + (−29.1 + 21.1i)6-s + 11.3·7-s + (51.7 − 37.6i)8-s + (25.0 + 77.0i)9-s + (223. + 15.8i)10-s + (−11.0 + 33.9i)11-s + (−44.4 − 136. i)12-s + (18.6 + 57.2i)13-s + (−14.0 + 43.1i)14-s + (188. − 466. i)15-s + (79.1 + 243. i)16-s + (−1.22e3 + 889. i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 + 0.672i)2-s + (0.467 + 0.339i)3-s + (−0.404 − 0.293i)4-s + (−0.240 − 0.970i)5-s + (−0.330 + 0.239i)6-s + 0.0874·7-s + (0.286 − 0.207i)8-s + (0.103 + 0.317i)9-s + (0.705 + 0.0500i)10-s + (−0.0274 + 0.0845i)11-s + (−0.0892 − 0.274i)12-s + (0.0305 + 0.0939i)13-s + (−0.0191 + 0.0588i)14-s + (0.216 − 0.535i)15-s + (0.0772 + 0.237i)16-s + (−1.02 + 0.746i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.109i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.5173969609\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5173969609\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.23 - 3.80i)T \) |
| 3 | \( 1 + (-7.28 - 5.29i)T \) |
| 5 | \( 1 + (13.4 + 54.2i)T \) |
| good | 7 | \( 1 - 11.3T + 1.68e4T^{2} \) |
| 11 | \( 1 + (11.0 - 33.9i)T + (-1.30e5 - 9.46e4i)T^{2} \) |
| 13 | \( 1 + (-18.6 - 57.2i)T + (-3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (1.22e3 - 889. i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (927. - 674. i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + (759. - 2.33e3i)T + (-5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (1.79e3 + 1.30e3i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (811. - 589. i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-1.91e3 - 5.89e3i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (1.33e3 + 4.09e3i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 - 1.30e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (1.12e4 + 8.19e3i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-4.33e3 - 3.15e3i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (7.96e3 + 2.45e4i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (1.54e4 - 4.75e4i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + (2.61e4 - 1.90e4i)T + (4.17e8 - 1.28e9i)T^{2} \) |
| 71 | \( 1 + (4.39e4 + 3.19e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-13.6 + 41.9i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-1.86e4 - 1.35e4i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-5.35e4 + 3.89e4i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + (4.39e3 - 1.35e4i)T + (-4.51e9 - 3.28e9i)T^{2} \) |
| 97 | \( 1 + (3.81e3 + 2.77e3i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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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
−12.90855106929059897569264050795, −11.63335250572530267422861446225, −10.33307876549691059164070147665, −9.242839542710470744507708983889, −8.498728601285221475545713711705, −7.62370171243353078588752530011, −6.14917409649189740163623980601, −4.86649899225762429211142323382, −3.86584310711020458606508079580, −1.70271281536966004065178886449,
0.16884145221210801293045733892, 2.09542084367625079331670980158, 3.10455185170880874386705663111, 4.45792481454357222763073436894, 6.38859932353061505239788206633, 7.42967397524686660282036530381, 8.519723478670146900970843456840, 9.570940165732118636162196512608, 10.74472767220872751014238696214, 11.39714322736159244596431426344
