| 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
−14.62449250238069714131656001442, −13.50885674141312808901423890982, −12.91558865493989340481260244342, −11.21768530001349410731534443219, −9.944561350812310523082454841276, −8.831033607623491023447750484050, −8.405529455125938588703523578228, −6.25188232002687829578721762998, −4.87686350799744976064480303019, −2.93222456547653746812145090168,
1.81654755443271762474915912942, 4.42349753045772328140955406688, 6.05596560020222241825192065492, 7.50762091212198572927666738715, 8.823030111367120039838882948622, 10.07946016674208063068082597898, 10.51303457145563669012591896224, 12.60407754797607785150802970544, 13.40033547481754246126628911705, 14.24425118602731108508175003714
