| L(s) = 1 | + (−1.38 + 7.87i)2-s + (−66.5 − 55.8i)3-s + (−60.1 − 21.8i)4-s + (−312. + 113. i)5-s + (532. − 446. i)6-s + (−817. − 1.41e3i)7-s + (256 − 443. i)8-s + (930. + 5.27e3i)9-s + (−462. − 2.62e3i)10-s + (−853. + 1.47e3i)11-s + (2.77e3 + 4.81e3i)12-s + (4.23e3 − 3.55e3i)13-s + (1.22e4 − 4.47e3i)14-s + (2.71e4 + 9.88e3i)15-s + (3.13e3 + 2.63e3i)16-s + (−3.80e3 + 2.16e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 + 0.696i)2-s + (−1.42 − 1.19i)3-s + (−0.469 − 0.171i)4-s + (−1.11 + 0.407i)5-s + (1.00 − 0.844i)6-s + (−0.901 − 1.56i)7-s + (0.176 − 0.306i)8-s + (0.425 + 2.41i)9-s + (−0.146 − 0.828i)10-s + (−0.193 + 0.335i)11-s + (0.464 + 0.804i)12-s + (0.535 − 0.449i)13-s + (1.19 − 0.435i)14-s + (2.07 + 0.756i)15-s + (0.191 + 0.160i)16-s + (−0.188 + 1.06i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.490 - 0.871i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.490 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.215081 + 0.125738i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.215081 + 0.125738i\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + (1.56e4 + 2.54e4i)T \) |
| good | 3 | \( 1 + (66.5 + 55.8i)T + (379. + 2.15e3i)T^{2} \) |
| 5 | \( 1 + (312. - 113. i)T + (5.98e4 - 5.02e4i)T^{2} \) |
| 7 | \( 1 + (817. + 1.41e3i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (853. - 1.47e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-4.23e3 + 3.55e3i)T + (1.08e7 - 6.17e7i)T^{2} \) |
| 17 | \( 1 + (3.80e3 - 2.16e4i)T + (-3.85e8 - 1.40e8i)T^{2} \) |
| 23 | \( 1 + (-6.83e3 - 2.48e3i)T + (2.60e9 + 2.18e9i)T^{2} \) |
| 29 | \( 1 + (1.06e3 + 6.05e3i)T + (-1.62e10 + 5.89e9i)T^{2} \) |
| 31 | \( 1 + (1.00e5 + 1.74e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 - 5.86e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (2.97e5 + 2.49e5i)T + (3.38e10 + 1.91e11i)T^{2} \) |
| 43 | \( 1 + (1.07e5 - 3.92e4i)T + (2.08e11 - 1.74e11i)T^{2} \) |
| 47 | \( 1 + (-1.29e4 - 7.33e4i)T + (-4.76e11 + 1.73e11i)T^{2} \) |
| 53 | \( 1 + (-1.45e5 - 5.31e4i)T + (8.99e11 + 7.55e11i)T^{2} \) |
| 59 | \( 1 + (4.40e5 - 2.50e6i)T + (-2.33e12 - 8.51e11i)T^{2} \) |
| 61 | \( 1 + (-6.26e4 - 2.28e4i)T + (2.40e12 + 2.02e12i)T^{2} \) |
| 67 | \( 1 + (-1.18e5 - 6.69e5i)T + (-5.69e12 + 2.07e12i)T^{2} \) |
| 71 | \( 1 + (1.84e6 - 6.72e5i)T + (6.96e12 - 5.84e12i)T^{2} \) |
| 73 | \( 1 + (1.17e6 + 9.88e5i)T + (1.91e12 + 1.08e13i)T^{2} \) |
| 79 | \( 1 + (-4.53e6 - 3.80e6i)T + (3.33e12 + 1.89e13i)T^{2} \) |
| 83 | \( 1 + (-1.13e6 - 1.96e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (3.20e6 - 2.68e6i)T + (7.68e12 - 4.35e13i)T^{2} \) |
| 97 | \( 1 + (-9.20e5 + 5.22e6i)T + (-7.59e13 - 2.76e13i)T^{2} \) |
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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
−15.27351300714446059519777941523, −13.38832812446262970620966299002, −12.82594922056188521966167512828, −11.26921992695329650643689557118, −10.44343243820985276554925220041, −7.80945065629166026989475426956, −7.09361132542876183954103536442, −6.14888878101058307330679452913, −4.17858835236603316844955023150, −0.73077004691834179605014753982,
0.23131467958574997208636970142, 3.45105870412946885033445590449, 4.82762415375984513706336047274, 6.12705978795115581040197363258, 8.746545616056236345245157691598, 9.744496809232337374468556134220, 11.18054335286639490762455721030, 11.86570378425175739046141135688, 12.64134002526057872919198987485, 15.09316584168571485563597617920
