| L(s) = 1 | + (−0.173 − 0.984i)2-s + (−0.939 + 0.342i)4-s + (3.10 + 2.60i)5-s + (0.144 + 0.0525i)7-s + (0.5 + 0.866i)8-s + (2.02 − 3.50i)10-s + (−0.169 + 0.141i)11-s + (0.103 − 0.585i)13-s + (0.0266 − 0.151i)14-s + (0.766 − 0.642i)16-s + (2.78 − 4.81i)17-s + (−1.91 − 3.30i)19-s + (−3.80 − 1.38i)20-s + (0.169 + 0.141i)22-s + (−5.50 + 2.00i)23-s + ⋯ |
| L(s) = 1 | + (−0.122 − 0.696i)2-s + (−0.469 + 0.171i)4-s + (1.38 + 1.16i)5-s + (0.0545 + 0.0198i)7-s + (0.176 + 0.306i)8-s + (0.639 − 1.10i)10-s + (−0.0510 + 0.0428i)11-s + (0.0286 − 0.162i)13-s + (0.00712 − 0.0404i)14-s + (0.191 − 0.160i)16-s + (0.674 − 1.16i)17-s + (−0.438 − 0.759i)19-s + (−0.850 − 0.309i)20-s + (0.0360 + 0.0302i)22-s + (−1.14 + 0.417i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.298i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 + 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.21327 - 0.185270i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.21327 - 0.185270i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.173 + 0.984i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-3.10 - 2.60i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (-0.144 - 0.0525i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (0.169 - 0.141i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.103 + 0.585i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-2.78 + 4.81i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.91 + 3.30i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (5.50 - 2.00i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (0.129 + 0.736i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (4.77 - 1.73i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (1.87 - 3.24i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.690 - 3.91i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-7.81 + 6.56i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (0.447 + 0.162i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + 3.29T + 53T^{2} \) |
| 59 | \( 1 + (5.57 + 4.67i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (3.16 + 1.15i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.29 + 7.34i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-1.42 + 2.47i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.638 - 1.10i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.574 + 3.25i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.43 - 8.14i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (2.47 + 4.29i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.33 + 3.63i)T + (16.8 - 95.5i)T^{2} \) |
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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
−12.87887633974818993777436112019, −11.63095480284385394087645435167, −10.70613604924399111388440762026, −9.922616158429780300075898398561, −9.172680365043957052892026720816, −7.55549147083834327364876209280, −6.35993276732090414670837551305, −5.18260915768612758480293181738, −3.24725938870531440736237008463, −2.08072822401724539267379905701,
1.70213180785438036846964555543, 4.26456330794354388526997072344, 5.61606811388652442260099606933, 6.17249774563356720120609197277, 7.88503786318351217334084860213, 8.795618713096398899742074332967, 9.693807658905118583538850002673, 10.54795009992386494575519398667, 12.39425732599848962977978075112, 12.91369439771208058994412682832
