| L(s) = 1 | + (−1.29 − 1.54i)2-s + (0.621 − 1.70i)3-s + (−0.00997 + 0.0565i)4-s + (1.24 + 7.05i)5-s + (−3.43 + 1.25i)6-s + (0.422 − 0.731i)7-s + (−6.87 + 3.97i)8-s + (4.36 + 3.66i)9-s + (9.27 − 11.0i)10-s + (−3.11 − 5.39i)11-s + (0.0903 + 0.0521i)12-s + (−5.42 − 14.8i)13-s + (−1.67 + 0.295i)14-s + (12.8 + 2.25i)15-s + (15.2 + 5.54i)16-s + (−19.3 + 16.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.647 − 0.771i)2-s + (0.207 − 0.568i)3-s + (−0.00249 + 0.0141i)4-s + (0.248 + 1.41i)5-s + (−0.572 + 0.208i)6-s + (0.0603 − 0.104i)7-s + (−0.859 + 0.496i)8-s + (0.485 + 0.407i)9-s + (0.927 − 1.10i)10-s + (−0.282 − 0.490i)11-s + (0.00753 + 0.00434i)12-s + (−0.417 − 1.14i)13-s + (−0.119 + 0.0211i)14-s + (0.854 + 0.150i)15-s + (0.952 + 0.346i)16-s + (−1.13 + 0.953i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.472 + 0.881i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.472 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.594531 - 0.355933i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.594531 - 0.355933i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (3.87 - 18.5i)T \) |
| good | 2 | \( 1 + (1.29 + 1.54i)T + (-0.694 + 3.93i)T^{2} \) |
| 3 | \( 1 + (-0.621 + 1.70i)T + (-6.89 - 5.78i)T^{2} \) |
| 5 | \( 1 + (-1.24 - 7.05i)T + (-23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (-0.422 + 0.731i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (3.11 + 5.39i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (5.42 + 14.8i)T + (-129. + 108. i)T^{2} \) |
| 17 | \( 1 + (19.3 - 16.2i)T + (50.1 - 284. i)T^{2} \) |
| 23 | \( 1 + (-6.55 + 37.1i)T + (-497. - 180. i)T^{2} \) |
| 29 | \( 1 + (-12.5 + 14.9i)T + (-146. - 828. i)T^{2} \) |
| 31 | \( 1 + (-8.08 - 4.66i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 1.20iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (4.05 - 11.1i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (4.58 + 25.9i)T + (-1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (-20.0 - 16.8i)T + (383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 + (22.4 + 3.95i)T + (2.63e3 + 960. i)T^{2} \) |
| 59 | \( 1 + (38.6 + 46.0i)T + (-604. + 3.42e3i)T^{2} \) |
| 61 | \( 1 + (-7.08 + 40.2i)T + (-3.49e3 - 1.27e3i)T^{2} \) |
| 67 | \( 1 + (41.8 - 49.8i)T + (-779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (-23.0 + 4.05i)T + (4.73e3 - 1.72e3i)T^{2} \) |
| 73 | \( 1 + (-62.4 - 22.7i)T + (4.08e3 + 3.42e3i)T^{2} \) |
| 79 | \( 1 + (25.5 - 70.1i)T + (-4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (5.25 - 9.09i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (9.14 + 25.1i)T + (-6.06e3 + 5.09e3i)T^{2} \) |
| 97 | \( 1 + (-107. - 128. i)T + (-1.63e3 + 9.26e3i)T^{2} \) |
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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
−18.53998368740312710370418416162, −17.50111503113355373326368748780, −15.31722653555601223269741153999, −14.20176218795446250624483642138, −12.65022133450786219681210371337, −10.77787581939024178268278685879, −10.31225340053894610562036829506, −8.164099986728773378132465545232, −6.38341205812640771042556486409, −2.51550721235901519973879287446,
4.70263772762855863523261366335, 7.06774014772897186226511350371, 8.952540956519586682832023849119, 9.441178135246443984840730657481, 11.96915965445523536740912694554, 13.29981724811582554635221045774, 15.32260566258806496553718141990, 16.06283824226466413082910615485, 17.12471921111084854547663063975, 18.08774631543861211184019535778
