| L(s) = 1 | + (−0.541 + 0.312i)2-s + (−8.70 + 5.02i)3-s + (−7.80 + 13.5i)4-s + (4.61 + 8.00i)5-s + (3.14 − 5.44i)6-s + 0.274·7-s − 19.7i·8-s + (10.0 − 17.4i)9-s + (−5.00 − 2.88i)10-s + 45.5·11-s − 156. i·12-s + (111. + 64.2i)13-s + (−0.148 + 0.0857i)14-s + (−80.4 − 46.4i)15-s + (−118. − 205. i)16-s + (212. + 367. i)17-s + ⋯ |
| L(s) = 1 | + (−0.135 + 0.0781i)2-s + (−0.967 + 0.558i)3-s + (−0.487 + 0.844i)4-s + (0.184 + 0.320i)5-s + (0.0873 − 0.151i)6-s + 0.00560·7-s − 0.308i·8-s + (0.124 − 0.215i)9-s + (−0.0500 − 0.0288i)10-s + 0.376·11-s − 1.09i·12-s + (0.658 + 0.380i)13-s + (−0.000758 + 0.000437i)14-s + (−0.357 − 0.206i)15-s + (−0.463 − 0.803i)16-s + (0.733 + 1.27i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.551 - 0.834i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.551 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.337455 + 0.627590i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.337455 + 0.627590i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (280. - 227. i)T \) |
| good | 2 | \( 1 + (0.541 - 0.312i)T + (8 - 13.8i)T^{2} \) |
| 3 | \( 1 + (8.70 - 5.02i)T + (40.5 - 70.1i)T^{2} \) |
| 5 | \( 1 + (-4.61 - 8.00i)T + (-312.5 + 541. i)T^{2} \) |
| 7 | \( 1 - 0.274T + 2.40e3T^{2} \) |
| 11 | \( 1 - 45.5T + 1.46e4T^{2} \) |
| 13 | \( 1 + (-111. - 64.2i)T + (1.42e4 + 2.47e4i)T^{2} \) |
| 17 | \( 1 + (-212. - 367. i)T + (-4.17e4 + 7.23e4i)T^{2} \) |
| 23 | \( 1 + (286. - 496. i)T + (-1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + (-763. - 440. i)T + (3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 + 1.53e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 71.9iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (-1.11e3 + 642. i)T + (1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (-603. - 1.04e3i)T + (-1.70e6 + 2.96e6i)T^{2} \) |
| 47 | \( 1 + (1.34e3 - 2.33e3i)T + (-2.43e6 - 4.22e6i)T^{2} \) |
| 53 | \( 1 + (-1.83e3 - 1.05e3i)T + (3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (-4.85e3 + 2.80e3i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (-1.24e3 + 2.14e3i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (656. + 379. i)T + (1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + (5.26e3 - 3.04e3i)T + (1.27e7 - 2.20e7i)T^{2} \) |
| 73 | \( 1 + (-3.51e3 - 6.08e3i)T + (-1.41e7 + 2.45e7i)T^{2} \) |
| 79 | \( 1 + (2.79e3 - 1.61e3i)T + (1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 - 5.58e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + (4.91e3 + 2.83e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 + (-1.02e4 + 5.93e3i)T + (4.42e7 - 7.66e7i)T^{2} \) |
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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
−17.80817911578294351380588103280, −16.95581828868615789478481353519, −16.06331681203195646907527130826, −14.28033405934703895912574761059, −12.69529520412606049629464123943, −11.37808492801859042467158126296, −9.992295858441862209492341537482, −8.226063404039635843470417072329, −6.11561328951823225836511385998, −4.09011585580201676911616834321,
0.825526933680763760019763622280, 5.15355865318842038159412029035, 6.52234829806753048460194540270, 8.883073701908276963435688326710, 10.48098090784681275412628949196, 11.83494496355638514429726637952, 13.22932203245911104983714738422, 14.57826461886475648812197836943, 16.26675796089847711546793691276, 17.62763296479661260174160565919
