| L(s) = 1 | + (−1.73 − 4.77i)2-s + (14.9 + 2.63i)3-s + (−7.55 + 6.33i)4-s + (−20.3 − 17.0i)5-s + (−13.3 − 75.9i)6-s + (19.4 + 33.7i)7-s + (−27.0 − 15.6i)8-s + (139. + 50.9i)9-s + (−46.2 + 126. i)10-s + (−79.5 + 137. i)11-s + (−129. + 74.7i)12-s + (211. − 37.2i)13-s + (127. − 151. i)14-s + (−258. − 308. i)15-s + (−54.9 + 311. i)16-s + (−29.5 + 10.7i)17-s + ⋯ |
| L(s) = 1 | + (−0.434 − 1.19i)2-s + (1.65 + 0.292i)3-s + (−0.472 + 0.396i)4-s + (−0.813 − 0.682i)5-s + (−0.371 − 2.10i)6-s + (0.397 + 0.688i)7-s + (−0.422 − 0.243i)8-s + (1.72 + 0.628i)9-s + (−0.462 + 1.26i)10-s + (−0.657 + 1.13i)11-s + (−0.899 + 0.519i)12-s + (1.24 − 0.220i)13-s + (0.649 − 0.773i)14-s + (−1.15 − 1.37i)15-s + (−0.214 + 1.21i)16-s + (−0.102 + 0.0371i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.219 + 0.975i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.219 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.12878 - 0.902690i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12878 - 0.902690i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (347. - 98.3i)T \) |
| good | 2 | \( 1 + (1.73 + 4.77i)T + (-12.2 + 10.2i)T^{2} \) |
| 3 | \( 1 + (-14.9 - 2.63i)T + (76.1 + 27.7i)T^{2} \) |
| 5 | \( 1 + (20.3 + 17.0i)T + (108. + 615. i)T^{2} \) |
| 7 | \( 1 + (-19.4 - 33.7i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (79.5 - 137. i)T + (-7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-211. + 37.2i)T + (2.68e4 - 9.76e3i)T^{2} \) |
| 17 | \( 1 + (29.5 - 10.7i)T + (6.39e4 - 5.36e4i)T^{2} \) |
| 23 | \( 1 + (47.0 - 39.4i)T + (4.85e4 - 2.75e5i)T^{2} \) |
| 29 | \( 1 + (-502. + 1.38e3i)T + (-5.41e5 - 4.54e5i)T^{2} \) |
| 31 | \( 1 + (751. - 433. i)T + (4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + 814. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (1.05e3 + 185. i)T + (2.65e6 + 9.66e5i)T^{2} \) |
| 43 | \( 1 + (1.28e3 + 1.07e3i)T + (5.93e5 + 3.36e6i)T^{2} \) |
| 47 | \( 1 + (-577. - 210. i)T + (3.73e6 + 3.13e6i)T^{2} \) |
| 53 | \( 1 + (-1.68e3 - 2.01e3i)T + (-1.37e6 + 7.77e6i)T^{2} \) |
| 59 | \( 1 + (-984. - 2.70e3i)T + (-9.28e6 + 7.78e6i)T^{2} \) |
| 61 | \( 1 + (-893. + 749. i)T + (2.40e6 - 1.36e7i)T^{2} \) |
| 67 | \( 1 + (794. - 2.18e3i)T + (-1.54e7 - 1.29e7i)T^{2} \) |
| 71 | \( 1 + (431. - 513. i)T + (-4.41e6 - 2.50e7i)T^{2} \) |
| 73 | \( 1 + (-396. + 2.24e3i)T + (-2.66e7 - 9.71e6i)T^{2} \) |
| 79 | \( 1 + (-8.30e3 - 1.46e3i)T + (3.66e7 + 1.33e7i)T^{2} \) |
| 83 | \( 1 + (1.13e3 + 1.96e3i)T + (-2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-3.76e3 + 664. i)T + (5.89e7 - 2.14e7i)T^{2} \) |
| 97 | \( 1 + (-4.05e3 - 1.11e4i)T + (-6.78e7 + 5.69e7i)T^{2} \) |
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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
−18.16109516105645834448458775339, −15.67636595879585875069436658775, −15.08582117181646198992631447724, −13.20708017289567605344119534386, −12.07269699865263428071379211013, −10.35630481707712650424384535435, −8.940137078155028812375866529732, −8.147321037648184323954011928466, −3.99150576202275262923552569262, −2.17144416926335349310521181307,
3.37563619187788746140586810594, 6.84147879868949255699843182646, 8.015838424343407502282104940135, 8.691715295062395341108347457528, 10.97878202073945386894829345106, 13.42269395076460339344915324825, 14.44551530213967616454850451304, 15.34604987173954006644908781957, 16.37569938689627670799193024137, 18.25360903805869959387960346238
