| L(s) = 1 | + (1.81 + 2.16i)2-s + (−0.486 + 1.33i)3-s + (−1.38 + 7.87i)4-s + (3.87 + 21.9i)5-s + (−3.77 + 1.37i)6-s + (−12.4 + 21.6i)7-s + (−19.5 + 11.3i)8-s + (60.5 + 50.7i)9-s + (−40.5 + 48.3i)10-s + (−23.8 − 41.2i)11-s + (−9.85 − 5.68i)12-s + (−16.3 − 44.9i)13-s + (−69.5 + 12.2i)14-s + (−31.2 − 5.50i)15-s + (−60.1 − 21.8i)16-s + (200. − 168. i)17-s + ⋯ |
| L(s) = 1 | + (0.454 + 0.541i)2-s + (−0.0540 + 0.148i)3-s + (−0.0868 + 0.492i)4-s + (0.154 + 0.878i)5-s + (−0.104 + 0.0382i)6-s + (−0.254 + 0.441i)7-s + (−0.306 + 0.176i)8-s + (0.746 + 0.626i)9-s + (−0.405 + 0.483i)10-s + (−0.196 − 0.340i)11-s + (−0.0684 − 0.0395i)12-s + (−0.0967 − 0.265i)13-s + (−0.354 + 0.0625i)14-s + (−0.138 − 0.0244i)15-s + (−0.234 − 0.0855i)16-s + (0.694 − 0.582i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.145 - 0.989i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.145 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.15333 + 1.33525i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15333 + 1.33525i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.81 - 2.16i)T \) |
| 19 | \( 1 + (-357. + 52.7i)T \) |
| good | 3 | \( 1 + (0.486 - 1.33i)T + (-62.0 - 52.0i)T^{2} \) |
| 5 | \( 1 + (-3.87 - 21.9i)T + (-587. + 213. i)T^{2} \) |
| 7 | \( 1 + (12.4 - 21.6i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (23.8 + 41.2i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (16.3 + 44.9i)T + (-2.18e4 + 1.83e4i)T^{2} \) |
| 17 | \( 1 + (-200. + 168. i)T + (1.45e4 - 8.22e4i)T^{2} \) |
| 23 | \( 1 + (-108. + 617. i)T + (-2.62e5 - 9.57e4i)T^{2} \) |
| 29 | \( 1 + (115. - 137. i)T + (-1.22e5 - 6.96e5i)T^{2} \) |
| 31 | \( 1 + (307. + 177. i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 - 596. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (862. - 2.36e3i)T + (-2.16e6 - 1.81e6i)T^{2} \) |
| 43 | \( 1 + (-103. - 587. i)T + (-3.21e6 + 1.16e6i)T^{2} \) |
| 47 | \( 1 + (627. + 526. i)T + (8.47e5 + 4.80e6i)T^{2} \) |
| 53 | \( 1 + (-468. - 82.6i)T + (7.41e6 + 2.69e6i)T^{2} \) |
| 59 | \( 1 + (3.81e3 + 4.54e3i)T + (-2.10e6 + 1.19e7i)T^{2} \) |
| 61 | \( 1 + (-876. + 4.96e3i)T + (-1.30e7 - 4.73e6i)T^{2} \) |
| 67 | \( 1 + (-3.68e3 + 4.39e3i)T + (-3.49e6 - 1.98e7i)T^{2} \) |
| 71 | \( 1 + (4.61e3 - 813. i)T + (2.38e7 - 8.69e6i)T^{2} \) |
| 73 | \( 1 + (-5.01e3 - 1.82e3i)T + (2.17e7 + 1.82e7i)T^{2} \) |
| 79 | \( 1 + (-2.86e3 + 7.87e3i)T + (-2.98e7 - 2.50e7i)T^{2} \) |
| 83 | \( 1 + (1.94e3 - 3.36e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-2.77e3 - 7.61e3i)T + (-4.80e7 + 4.03e7i)T^{2} \) |
| 97 | \( 1 + (5.99e3 + 7.14e3i)T + (-1.53e7 + 8.71e7i)T^{2} \) |
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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
−15.80962613581664272771277877069, −14.72548633240468219153967178406, −13.69472523053871460822255395303, −12.48608480533161607745129017363, −10.99150508655311728252649459326, −9.700413883793495420231660923265, −7.87103770880083655080813674302, −6.59955606209514646499043208356, −5.06709389452783407111590291006, −2.99967973412891416768095645819,
1.25952176928942921497464225026, 3.83147256345333660917663264728, 5.43242091913278373097428973202, 7.25933813300947510380209952881, 9.206821824708824544488079589832, 10.23879002862412020606329975604, 11.92104909484709478980657996300, 12.76771759363754836128711315275, 13.72024421862905192777040231932, 15.15267185021297105145677410238
