| 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
−13.09141152437444273534513593059, −11.53122170117696835452335432585, −11.02134712182742584899023254783, −9.630579314596080491614824042307, −8.280667934635139323686392815404, −7.74398130869700858752961078828, −6.24702801392860837317614244493, −5.19877868720360795000380287849, −4.14405237065493428086901073087, −1.95374505196727808964199595728,
1.29866338711975834860675872586, 2.91930591466342271255665843281, 4.62984801417010884757572277639, 5.54332882492865171980114377533, 6.96863160044273984907823606668, 8.639090746345531196852201320936, 9.127581246692503867254563246870, 10.72269385116156301481313058054, 11.27134692629598098828204571505, 12.23908201386063078763213804813
