| L(s) = 1 | + (−0.0747 − 0.997i)2-s + (0.702 + 0.216i)3-s + (−0.988 + 0.149i)4-s + (2.03 − 1.88i)5-s + (0.163 − 0.717i)6-s + (0.222 + 0.974i)8-s + (−2.03 − 1.38i)9-s + (−2.03 − 1.88i)10-s + (2.29 − 1.56i)11-s + (−0.727 − 0.109i)12-s + (−3.14 − 1.51i)13-s + (1.83 − 0.885i)15-s + (0.955 − 0.294i)16-s + (−1.60 − 4.08i)17-s + (−1.22 + 2.12i)18-s + (2.26 + 3.92i)19-s + ⋯ |
| L(s) = 1 | + (−0.0528 − 0.705i)2-s + (0.405 + 0.125i)3-s + (−0.494 + 0.0745i)4-s + (0.909 − 0.843i)5-s + (0.0668 − 0.292i)6-s + (0.0786 + 0.344i)8-s + (−0.677 − 0.461i)9-s + (−0.642 − 0.596i)10-s + (0.691 − 0.471i)11-s + (−0.209 − 0.0316i)12-s + (−0.872 − 0.419i)13-s + (0.474 − 0.228i)15-s + (0.238 − 0.0736i)16-s + (−0.389 − 0.991i)17-s + (−0.289 + 0.501i)18-s + (0.520 + 0.901i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 686 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.456 + 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 686 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.456 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.864338 - 1.41432i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.864338 - 1.41432i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0747 + 0.997i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (-0.702 - 0.216i)T + (2.47 + 1.68i)T^{2} \) |
| 5 | \( 1 + (-2.03 + 1.88i)T + (0.373 - 4.98i)T^{2} \) |
| 11 | \( 1 + (-2.29 + 1.56i)T + (4.01 - 10.2i)T^{2} \) |
| 13 | \( 1 + (3.14 + 1.51i)T + (8.10 + 10.1i)T^{2} \) |
| 17 | \( 1 + (1.60 + 4.08i)T + (-12.4 + 11.5i)T^{2} \) |
| 19 | \( 1 + (-2.26 - 3.92i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.17 + 2.99i)T + (-16.8 - 15.6i)T^{2} \) |
| 29 | \( 1 + (-5.83 + 7.31i)T + (-6.45 - 28.2i)T^{2} \) |
| 31 | \( 1 + (4.56 - 7.91i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-8.28 - 1.24i)T + (35.3 + 10.9i)T^{2} \) |
| 41 | \( 1 + (-0.176 - 0.773i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (-0.242 + 1.06i)T + (-38.7 - 18.6i)T^{2} \) |
| 47 | \( 1 + (0.217 + 2.90i)T + (-46.4 + 7.00i)T^{2} \) |
| 53 | \( 1 + (8.25 - 1.24i)T + (50.6 - 15.6i)T^{2} \) |
| 59 | \( 1 + (2.47 + 2.29i)T + (4.40 + 58.8i)T^{2} \) |
| 61 | \( 1 + (0.390 + 0.0588i)T + (58.2 + 17.9i)T^{2} \) |
| 67 | \( 1 + (-5.94 + 10.2i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.55 - 5.71i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (0.685 - 9.14i)T + (-72.1 - 10.8i)T^{2} \) |
| 79 | \( 1 + (-2.66 - 4.62i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-9.31 + 4.48i)T + (51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (1.37 + 0.937i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.880924699157113061974274665904, −9.473296232698663068546640851939, −8.756094790554749538603793933269, −7.944712937669051413545202118451, −6.47826536990563816728743657545, −5.49848518719957116334910720365, −4.63481649942670954036707206127, −3.32892123886709761546395126141, −2.34166926725775264580517458331, −0.873291370787342466590905866163,
1.94892169420838202661759587248, 3.00581109516612576064611612091, 4.46445034171997412982132544087, 5.55569898870947664083658279448, 6.45217007794663974884390369720, 7.15402200390892417739701988746, 8.028947265401387302819073166920, 9.195099697700742074063110059267, 9.553440483887301661423485408455, 10.66171129129589604679445572450
