| L(s) = 1 | + (−0.939 + 0.342i)2-s + (0.158 − 1.72i)3-s + (0.766 − 0.642i)4-s + (0.949 + 2.60i)5-s + (0.440 + 1.67i)6-s − 4.55·7-s + (−0.500 + 0.866i)8-s + (−2.94 − 0.548i)9-s + (−1.78 − 2.12i)10-s + (−3.04 + 1.75i)11-s + (−0.986 − 1.42i)12-s + (−4.41 + 0.779i)13-s + (4.27 − 1.55i)14-s + (4.65 − 1.22i)15-s + (0.173 − 0.984i)16-s + (3.85 − 4.59i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 + 0.241i)2-s + (0.0917 − 0.995i)3-s + (0.383 − 0.321i)4-s + (0.424 + 1.16i)5-s + (0.179 + 0.683i)6-s − 1.72·7-s + (−0.176 + 0.306i)8-s + (−0.983 − 0.182i)9-s + (−0.564 − 0.672i)10-s + (−0.917 + 0.529i)11-s + (−0.284 − 0.410i)12-s + (−1.22 + 0.216i)13-s + (1.14 − 0.416i)14-s + (1.20 − 0.315i)15-s + (0.0434 − 0.246i)16-s + (0.935 − 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.898 - 0.439i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.898 - 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0359770 + 0.155401i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0359770 + 0.155401i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.939 - 0.342i)T \) |
| 3 | \( 1 + (-0.158 + 1.72i)T \) |
| 19 | \( 1 + (1.17 - 4.19i)T \) |
| good | 5 | \( 1 + (-0.949 - 2.60i)T + (-3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + 4.55T + 7T^{2} \) |
| 11 | \( 1 + (3.04 - 1.75i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (4.41 - 0.779i)T + (12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-3.85 + 4.59i)T + (-2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (3.19 + 3.80i)T + (-3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-0.969 - 5.49i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (5.63 + 3.25i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 3.99iT - 37T^{2} \) |
| 41 | \( 1 + (-3.05 - 2.56i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (3.05 + 2.56i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-4.73 + 0.835i)T + (44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-0.764 - 4.33i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (0.992 - 5.62i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (9.13 + 3.32i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-4.59 + 12.6i)T + (-51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (0.450 - 2.55i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-0.318 + 0.115i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-3.59 - 0.633i)T + (74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 - 13.6iT - 83T^{2} \) |
| 89 | \( 1 + (0.752 + 0.273i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (1.72 + 4.72i)T + (-74.3 + 62.3i)T^{2} \) |
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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
−12.18094834671221927108879241470, −10.67876250831057367588311727130, −9.980335133268386360029642383871, −9.347010665012264430972966899412, −7.81780025215089552775823815198, −7.12744867048539096584769673754, −6.51309127206751213203174494952, −5.56042578042378194350312584939, −3.07695262688682264033081162129, −2.36822391498715516932752624511,
0.12169645950455567816208616074, 2.62409572951540373135152640194, 3.75758198562176079006512151228, 5.23667472534972235408534067416, 6.03180816472564420680586232841, 7.62196915251964441063397617304, 8.697601673604820282206869761050, 9.462243043175775583763263889124, 9.965415154468248274097334767281, 10.71336443542373637476740280804
