| L(s) = 1 | + (0.824 + 4.67i)2-s + (5.76 − 4.83i)3-s + (−13.6 + 4.98i)4-s + (−14.0 − 5.10i)5-s + (27.3 + 22.9i)6-s + (3.64 − 6.31i)7-s + (−15.6 − 27.0i)8-s + (5.12 − 29.0i)9-s + (12.3 − 69.8i)10-s + (28.9 + 50.1i)11-s + (−54.7 + 94.8i)12-s + (−23.4 − 19.6i)13-s + (32.5 + 11.8i)14-s + (−105. + 38.3i)15-s + (24.2 − 20.3i)16-s + (3.85 + 21.8i)17-s + ⋯ |
| L(s) = 1 | + (0.291 + 1.65i)2-s + (1.10 − 0.930i)3-s + (−1.71 + 0.622i)4-s + (−1.25 − 0.456i)5-s + (1.86 + 1.56i)6-s + (0.196 − 0.340i)7-s + (−0.689 − 1.19i)8-s + (0.189 − 1.07i)9-s + (0.389 − 2.20i)10-s + (0.793 + 1.37i)11-s + (−1.31 + 2.28i)12-s + (−0.499 − 0.419i)13-s + (0.621 + 0.226i)14-s + (−1.81 + 0.660i)15-s + (0.379 − 0.318i)16-s + (0.0549 + 0.311i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.359 - 0.933i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.359 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.08794 + 0.746464i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.08794 + 0.746464i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (-2.88 + 82.7i)T \) |
| good | 2 | \( 1 + (-0.824 - 4.67i)T + (-7.51 + 2.73i)T^{2} \) |
| 3 | \( 1 + (-5.76 + 4.83i)T + (4.68 - 26.5i)T^{2} \) |
| 5 | \( 1 + (14.0 + 5.10i)T + (95.7 + 80.3i)T^{2} \) |
| 7 | \( 1 + (-3.64 + 6.31i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-28.9 - 50.1i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (23.4 + 19.6i)T + (381. + 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-3.85 - 21.8i)T + (-4.61e3 + 1.68e3i)T^{2} \) |
| 23 | \( 1 + (3.80 - 1.38i)T + (9.32e3 - 7.82e3i)T^{2} \) |
| 29 | \( 1 + (15.0 - 85.1i)T + (-2.29e4 - 8.34e3i)T^{2} \) |
| 31 | \( 1 + (-54.4 + 94.3i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 - 158.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-101. + 84.9i)T + (1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + (228. + 83.0i)T + (6.09e4 + 5.11e4i)T^{2} \) |
| 47 | \( 1 + (-44.3 + 251. i)T + (-9.75e4 - 3.55e4i)T^{2} \) |
| 53 | \( 1 + (-120. + 43.9i)T + (1.14e5 - 9.56e4i)T^{2} \) |
| 59 | \( 1 + (-130. - 738. i)T + (-1.92e5 + 7.02e4i)T^{2} \) |
| 61 | \( 1 + (582. - 211. i)T + (1.73e5 - 1.45e5i)T^{2} \) |
| 67 | \( 1 + (-18.4 + 104. i)T + (-2.82e5 - 1.02e5i)T^{2} \) |
| 71 | \( 1 + (1.04e3 + 381. i)T + (2.74e5 + 2.30e5i)T^{2} \) |
| 73 | \( 1 + (-587. + 492. i)T + (6.75e4 - 3.83e5i)T^{2} \) |
| 79 | \( 1 + (42.1 - 35.3i)T + (8.56e4 - 4.85e5i)T^{2} \) |
| 83 | \( 1 + (77.4 - 134. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (617. + 518. i)T + (1.22e5 + 6.94e5i)T^{2} \) |
| 97 | \( 1 + (25.6 + 145. i)T + (-8.57e5 + 3.12e5i)T^{2} \) |
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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
−18.00047348559627673686483519101, −16.83572259781056476588680753984, −15.30425420312976504704934017861, −14.74585774464930676846998082268, −13.41510527095279214643140280109, −12.25015251753823355675548231621, −8.922317144352329385486730601750, −7.73290620206485525092770501635, −7.09490072640516785642662834080, −4.39650632406588305967414314283,
3.12469922670238722752503231938, 4.13705596117577828116897284437, 8.364147702628140400791917509095, 9.649205301203317854584770289427, 11.11006980051457222991261058371, 12.00276066907401681083882974551, 13.93164948359646993828488760943, 14.78623964527869869426042587399, 16.19362287317923906457571736510, 18.76997999828082017582765180814
