| L(s) = 1 | + (−0.510 − 0.165i)2-s + (0.587 − 0.809i)3-s + (−1.38 − 1.00i)4-s + (2.22 + 0.187i)5-s + (−0.434 + 0.315i)6-s − 2.57i·7-s + (1.17 + 1.61i)8-s + (−0.309 − 0.951i)9-s + (−1.10 − 0.465i)10-s + (−1.58 + 4.89i)11-s + (−1.62 + 0.529i)12-s + (1.40 − 0.455i)13-s + (−0.426 + 1.31i)14-s + (1.46 − 1.69i)15-s + (0.727 + 2.24i)16-s + (0.404 + 0.556i)17-s + ⋯ |
| L(s) = 1 | + (−0.360 − 0.117i)2-s + (0.339 − 0.467i)3-s + (−0.692 − 0.503i)4-s + (0.996 + 0.0838i)5-s + (−0.177 + 0.128i)6-s − 0.972i·7-s + (0.413 + 0.569i)8-s + (−0.103 − 0.317i)9-s + (−0.349 − 0.147i)10-s + (−0.479 + 1.47i)11-s + (−0.470 + 0.152i)12-s + (0.389 − 0.126i)13-s + (−0.114 + 0.350i)14-s + (0.377 − 0.436i)15-s + (0.181 + 0.560i)16-s + (0.0980 + 0.134i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.653 + 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.653 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.773552 - 0.354298i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.773552 - 0.354298i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.587 + 0.809i)T \) |
| 5 | \( 1 + (-2.22 - 0.187i)T \) |
| good | 2 | \( 1 + (0.510 + 0.165i)T + (1.61 + 1.17i)T^{2} \) |
| 7 | \( 1 + 2.57iT - 7T^{2} \) |
| 11 | \( 1 + (1.58 - 4.89i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (-1.40 + 0.455i)T + (10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.404 - 0.556i)T + (-5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (6.54 - 4.75i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-0.354 - 0.115i)T + (18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-0.0288 - 0.0209i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-3.63 + 2.63i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.81 + 0.590i)T + (29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (1.59 + 4.91i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 11.4iT - 43T^{2} \) |
| 47 | \( 1 + (5.00 - 6.89i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (5.36 - 7.38i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.0544 - 0.167i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (1.98 - 6.09i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (-0.0490 - 0.0675i)T + (-20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (9.83 + 7.14i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (11.4 + 3.72i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-4.01 - 2.91i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (5.50 + 7.57i)T + (-25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-0.00380 + 0.0117i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-4.47 + 6.16i)T + (-29.9 - 92.2i)T^{2} \) |
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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
−14.24425118602731108508175003714, −13.40033547481754246126628911705, −12.60407754797607785150802970544, −10.51303457145563669012591896224, −10.07946016674208063068082597898, −8.823030111367120039838882948622, −7.50762091212198572927666738715, −6.05596560020222241825192065492, −4.42349753045772328140955406688, −1.81654755443271762474915912942,
2.93222456547653746812145090168, 4.87686350799744976064480303019, 6.25188232002687829578721762998, 8.405529455125938588703523578228, 8.831033607623491023447750484050, 9.944561350812310523082454841276, 11.21768530001349410731534443219, 12.91558865493989340481260244342, 13.50885674141312808901423890982, 14.62449250238069714131656001442
