L(s) = 1 | + (−2.48 + 1.43i)3-s + (1.38 − 1.75i)5-s + 3.54i·7-s + (2.62 − 4.54i)9-s − 1.81·11-s + (2.78 + 1.60i)13-s + (−0.934 + 6.35i)15-s + (−6.92 + 3.99i)17-s + (0.863 + 4.27i)19-s + (−5.09 − 8.82i)21-s + (−7.30 − 4.21i)23-s + (−1.14 − 4.86i)25-s + 6.46i·27-s + (−4.29 + 7.43i)29-s − 1.70·31-s + ⋯ |
L(s) = 1 | + (−1.43 + 0.829i)3-s + (0.620 − 0.784i)5-s + 1.34i·7-s + (0.875 − 1.51i)9-s − 0.547·11-s + (0.771 + 0.445i)13-s + (−0.241 + 1.64i)15-s + (−1.67 + 0.969i)17-s + (0.198 + 0.980i)19-s + (−1.11 − 1.92i)21-s + (−1.52 − 0.878i)23-s + (−0.229 − 0.973i)25-s + 1.24i·27-s + (−0.796 + 1.38i)29-s − 0.306·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.775 - 0.631i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.197385 + 0.555425i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.197385 + 0.555425i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.38 + 1.75i)T \) |
| 19 | \( 1 + (-0.863 - 4.27i)T \) |
good | 3 | \( 1 + (2.48 - 1.43i)T + (1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 - 3.54iT - 7T^{2} \) |
| 11 | \( 1 + 1.81T + 11T^{2} \) |
| 13 | \( 1 + (-2.78 - 1.60i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (6.92 - 3.99i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (7.30 + 4.21i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.29 - 7.43i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 1.70T + 31T^{2} \) |
| 37 | \( 1 - 5.50iT - 37T^{2} \) |
| 41 | \( 1 + (-4.05 - 7.02i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.35 + 2.51i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.16 - 0.674i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.92 + 1.10i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.960 - 1.66i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.83 + 4.90i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-8.04 - 4.64i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.94 + 5.09i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.82 - 1.63i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.08 - 3.61i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 6.30iT - 83T^{2} \) |
| 89 | \( 1 + (-2.73 + 4.73i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.91 + 3.99i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.64720215031643004593738449228, −10.81866278360798483127604743173, −10.02189951175850464685755169300, −9.050568704729763441761427750557, −8.370043778414759129644435158363, −6.28516514858493415067293846072, −5.94801116657910300815463333597, −5.03104065290484363279162753819, −4.09302356425324325545397966114, −1.94259351280237042902666351107,
0.45237086967773660570062705820, 2.18720854172912261027715238931, 4.08418506058393439677131345912, 5.44573027839190507041563643093, 6.24966509590122256499212404953, 7.11698474663632002739729203687, 7.63857790969589912406501618673, 9.431915798212045909222888672698, 10.51897708577945155321286798029, 11.03722233607678918668325280961