| L(s) = 1 | + (−1.38 − 7.87i)2-s + (−33.6 + 28.2i)3-s + (−60.1 + 21.8i)4-s + (202. + 73.7i)5-s + (268. + 225. i)6-s + (234. − 406. i)7-s + (256 + 443. i)8-s + (−45.5 + 258. i)9-s + (299. − 1.69e3i)10-s + (−1.94e3 − 3.36e3i)11-s + (1.40e3 − 2.43e3i)12-s + (−7.91e3 − 6.63e3i)13-s + (−3.53e3 − 1.28e3i)14-s + (−8.89e3 + 3.23e3i)15-s + (3.13e3 − 2.63e3i)16-s + (−425. − 2.41e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 − 0.696i)2-s + (−0.718 + 0.603i)3-s + (−0.469 + 0.171i)4-s + (0.725 + 0.263i)5-s + (0.508 + 0.426i)6-s + (0.258 − 0.448i)7-s + (0.176 + 0.306i)8-s + (−0.0208 + 0.117i)9-s + (0.0947 − 0.537i)10-s + (−0.439 − 0.761i)11-s + (0.234 − 0.406i)12-s + (−0.998 − 0.838i)13-s + (−0.344 − 0.125i)14-s + (−0.680 + 0.247i)15-s + (0.191 − 0.160i)16-s + (−0.0210 − 0.119i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.864 + 0.502i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.864 + 0.502i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.149634 - 0.554907i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.149634 - 0.554907i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.38 + 7.87i)T \) |
| 19 | \( 1 + (2.57e4 + 1.52e4i)T \) |
| good | 3 | \( 1 + (33.6 - 28.2i)T + (379. - 2.15e3i)T^{2} \) |
| 5 | \( 1 + (-202. - 73.7i)T + (5.98e4 + 5.02e4i)T^{2} \) |
| 7 | \( 1 + (-234. + 406. i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + (1.94e3 + 3.36e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (7.91e3 + 6.63e3i)T + (1.08e7 + 6.17e7i)T^{2} \) |
| 17 | \( 1 + (425. + 2.41e3i)T + (-3.85e8 + 1.40e8i)T^{2} \) |
| 23 | \( 1 + (1.16e4 - 4.22e3i)T + (2.60e9 - 2.18e9i)T^{2} \) |
| 29 | \( 1 + (-9.65e3 + 5.47e4i)T + (-1.62e10 - 5.89e9i)T^{2} \) |
| 31 | \( 1 + (-1.31e5 + 2.27e5i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 - 3.31e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + (2.37e5 - 1.99e5i)T + (3.38e10 - 1.91e11i)T^{2} \) |
| 43 | \( 1 + (-1.80e5 - 6.58e4i)T + (2.08e11 + 1.74e11i)T^{2} \) |
| 47 | \( 1 + (-1.79e4 + 1.02e5i)T + (-4.76e11 - 1.73e11i)T^{2} \) |
| 53 | \( 1 + (8.58e5 - 3.12e5i)T + (8.99e11 - 7.55e11i)T^{2} \) |
| 59 | \( 1 + (3.30e5 + 1.87e6i)T + (-2.33e12 + 8.51e11i)T^{2} \) |
| 61 | \( 1 + (9.26e5 - 3.37e5i)T + (2.40e12 - 2.02e12i)T^{2} \) |
| 67 | \( 1 + (-5.55e4 + 3.15e5i)T + (-5.69e12 - 2.07e12i)T^{2} \) |
| 71 | \( 1 + (-1.46e6 - 5.33e5i)T + (6.96e12 + 5.84e12i)T^{2} \) |
| 73 | \( 1 + (3.74e6 - 3.14e6i)T + (1.91e12 - 1.08e13i)T^{2} \) |
| 79 | \( 1 + (4.48e6 - 3.76e6i)T + (3.33e12 - 1.89e13i)T^{2} \) |
| 83 | \( 1 + (-3.17e5 + 5.50e5i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 + (3.56e6 + 2.99e6i)T + (7.68e12 + 4.35e13i)T^{2} \) |
| 97 | \( 1 + (4.19e5 + 2.37e6i)T + (-7.59e13 + 2.76e13i)T^{2} \) |
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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
−14.06432772656387029147775891643, −13.04275742037374933397862461261, −11.49096285847000579076687719389, −10.54981394912060760346868313002, −9.810281685381584892494574112973, −7.980553503271270241243797607660, −5.87152238234566578179586980869, −4.57095682425853843312075752153, −2.51633963341867044483079511286, −0.27699707106528417178544197768,
1.78005497410191534931249087553, 4.88343935866952516943229978483, 6.09902165209255976105031916748, 7.23968644058146146479855061256, 8.895130926293598238781395516387, 10.15776545953086116641114856141, 11.96451462733843434474728419639, 12.79754999160574011733066725042, 14.19728985023818574908059346078, 15.26153717048437489143482551693
