| L(s) = 1 | + (−0.654 + 0.755i)2-s + (0.841 − 0.540i)3-s + (−0.142 − 0.989i)4-s + (0.415 + 0.909i)5-s + (−0.142 + 0.989i)6-s + (−1.69 + 0.496i)7-s + (0.841 + 0.540i)8-s + (0.415 − 0.909i)9-s + (−0.959 − 0.281i)10-s + (−3.55 − 4.09i)11-s + (−0.654 − 0.755i)12-s + (−6.78 − 1.99i)13-s + (0.731 − 1.60i)14-s + (0.841 + 0.540i)15-s + (−0.959 + 0.281i)16-s + (0.457 − 3.18i)17-s + ⋯ |
| L(s) = 1 | + (−0.463 + 0.534i)2-s + (0.485 − 0.312i)3-s + (−0.0711 − 0.494i)4-s + (0.185 + 0.406i)5-s + (−0.0580 + 0.404i)6-s + (−0.638 + 0.187i)7-s + (0.297 + 0.191i)8-s + (0.138 − 0.303i)9-s + (−0.303 − 0.0890i)10-s + (−1.07 − 1.23i)11-s + (−0.189 − 0.218i)12-s + (−1.88 − 0.552i)13-s + (0.195 − 0.428i)14-s + (0.217 + 0.139i)15-s + (−0.239 + 0.0704i)16-s + (0.110 − 0.771i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.472 + 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.472 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.242497 - 0.405244i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.242497 - 0.405244i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.654 - 0.755i)T \) |
| 3 | \( 1 + (-0.841 + 0.540i)T \) |
| 5 | \( 1 + (-0.415 - 0.909i)T \) |
| 23 | \( 1 + (4.70 + 0.928i)T \) |
| good | 7 | \( 1 + (1.69 - 0.496i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (3.55 + 4.09i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (6.78 + 1.99i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.457 + 3.18i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.185 - 1.29i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (0.0895 - 0.622i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-9.00 - 5.78i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-1.54 + 3.39i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (4.92 + 10.7i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (1.53 - 0.987i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 - 2.28T + 47T^{2} \) |
| 53 | \( 1 + (2.39 - 0.702i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (4.96 + 1.45i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (2.15 + 1.38i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (4.74 - 5.47i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (-8.17 + 9.42i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (0.615 + 4.27i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (-7.29 - 2.14i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (5.72 - 12.5i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (-7.86 + 5.05i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (3.20 + 7.01i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−10.06323743556749504571236356147, −9.332568573964446787706355300715, −8.269997545110809750487640851362, −7.65986842172816559721150174066, −6.80725527652635969095874571615, −5.84406531797146259153892474748, −4.96540331707795975830401732136, −3.17835573179019871578936479217, −2.43326153051928284569756480351, −0.24774068902082085821975988104,
2.00974359862943218341764639461, 2.83608170910425985698596087873, 4.31044049521697908283340264847, 4.94346474880050010213957258204, 6.50105970190262357658430605257, 7.60070839432349222946568967730, 8.131916350694016721140246756758, 9.407275157739130512316708697359, 9.969161896276207131342999183860, 10.18173585988081281611173113718
