| L(s) = 1 | + (0.541 − 1.30i)2-s + (−0.183 − 0.122i)3-s + (−1.41 − 1.41i)4-s + (2.19 − 0.436i)5-s + (−0.259 + 0.173i)6-s + (10.6 + 2.11i)7-s + (−2.61 + 1.08i)8-s + (−3.42 − 8.27i)9-s + (0.616 − 3.10i)10-s + (0.724 + 1.08i)11-s + (0.0859 + 0.432i)12-s + (6.85 − 6.85i)13-s + (8.50 − 12.7i)14-s + (−0.455 − 0.188i)15-s + 4i·16-s + (−16.8 − 1.87i)17-s + ⋯ |
| L(s) = 1 | + (0.270 − 0.653i)2-s + (−0.0610 − 0.0407i)3-s + (−0.353 − 0.353i)4-s + (0.438 − 0.0872i)5-s + (−0.0431 + 0.0288i)6-s + (1.51 + 0.301i)7-s + (−0.326 + 0.135i)8-s + (−0.380 − 0.918i)9-s + (0.0616 − 0.310i)10-s + (0.0658 + 0.0985i)11-s + (0.00716 + 0.0360i)12-s + (0.527 − 0.527i)13-s + (0.607 − 0.909i)14-s + (−0.0303 − 0.0125i)15-s + 0.250i·16-s + (−0.993 − 0.110i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.235 + 0.971i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.235 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.47890 - 1.16290i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.47890 - 1.16290i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.541 + 1.30i)T \) |
| 5 | \( 1 + (-2.19 + 0.436i)T \) |
| 17 | \( 1 + (16.8 + 1.87i)T \) |
| good | 3 | \( 1 + (0.183 + 0.122i)T + (3.44 + 8.31i)T^{2} \) |
| 7 | \( 1 + (-10.6 - 2.11i)T + (45.2 + 18.7i)T^{2} \) |
| 11 | \( 1 + (-0.724 - 1.08i)T + (-46.3 + 111. i)T^{2} \) |
| 13 | \( 1 + (-6.85 + 6.85i)T - 169iT^{2} \) |
| 19 | \( 1 + (-10.9 + 26.3i)T + (-255. - 255. i)T^{2} \) |
| 23 | \( 1 + (-7.65 + 5.11i)T + (202. - 488. i)T^{2} \) |
| 29 | \( 1 + (-10.2 - 51.3i)T + (-776. + 321. i)T^{2} \) |
| 31 | \( 1 + (23.8 - 35.7i)T + (-367. - 887. i)T^{2} \) |
| 37 | \( 1 + (-8.32 - 5.56i)T + (523. + 1.26e3i)T^{2} \) |
| 41 | \( 1 + (-38.8 - 7.72i)T + (1.55e3 + 643. i)T^{2} \) |
| 43 | \( 1 + (18.0 + 43.5i)T + (-1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 + (30.9 - 30.9i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (19.5 - 47.1i)T + (-1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (12.8 - 5.33i)T + (2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (21.2 - 106. i)T + (-3.43e3 - 1.42e3i)T^{2} \) |
| 67 | \( 1 - 14.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (-90.5 - 60.4i)T + (1.92e3 + 4.65e3i)T^{2} \) |
| 73 | \( 1 + (-5.53 + 1.10i)T + (4.92e3 - 2.03e3i)T^{2} \) |
| 79 | \( 1 + (75.3 + 112. i)T + (-2.38e3 + 5.76e3i)T^{2} \) |
| 83 | \( 1 + (87.7 + 36.3i)T + (4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-26.5 - 26.5i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (12.3 + 61.9i)T + (-8.69e3 + 3.60e3i)T^{2} \) |
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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
−12.23908201386063078763213804813, −11.27134692629598098828204571505, −10.72269385116156301481313058054, −9.127581246692503867254563246870, −8.639090746345531196852201320936, −6.96863160044273984907823606668, −5.54332882492865171980114377533, −4.62984801417010884757572277639, −2.91930591466342271255665843281, −1.29866338711975834860675872586,
1.95374505196727808964199595728, 4.14405237065493428086901073087, 5.19877868720360795000380287849, 6.24702801392860837317614244493, 7.74398130869700858752961078828, 8.280667934635139323686392815404, 9.630579314596080491614824042307, 11.02134712182742584899023254783, 11.53122170117696835452335432585, 13.09141152437444273534513593059
