| L(s) = 1 | + (−0.507 + 1.31i)2-s + (−0.540 − 0.841i)3-s + (−1.48 − 1.34i)4-s + (−0.282 + 0.128i)5-s + (1.38 − 0.286i)6-s + (−2.03 + 0.597i)7-s + (2.52 − 1.27i)8-s + (−0.415 + 0.909i)9-s + (−0.0267 − 0.438i)10-s + (2.44 − 2.12i)11-s + (−0.325 + 1.97i)12-s + (−0.746 + 2.54i)13-s + (0.244 − 2.99i)14-s + (0.261 + 0.167i)15-s + (0.405 + 3.97i)16-s + (−0.439 + 3.05i)17-s + ⋯ |
| L(s) = 1 | + (−0.359 + 0.933i)2-s + (−0.312 − 0.485i)3-s + (−0.742 − 0.670i)4-s + (−0.126 + 0.0576i)5-s + (0.565 − 0.116i)6-s + (−0.769 + 0.225i)7-s + (0.892 − 0.451i)8-s + (−0.138 + 0.303i)9-s + (−0.00847 − 0.138i)10-s + (0.738 − 0.639i)11-s + (−0.0939 + 0.569i)12-s + (−0.206 + 0.704i)13-s + (0.0654 − 0.799i)14-s + (0.0674 + 0.0433i)15-s + (0.101 + 0.994i)16-s + (−0.106 + 0.741i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.351 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.351 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.422501 + 0.609674i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.422501 + 0.609674i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.507 - 1.31i)T \) |
| 3 | \( 1 + (0.540 + 0.841i)T \) |
| 23 | \( 1 + (2.97 - 3.76i)T \) |
| good | 5 | \( 1 + (0.282 - 0.128i)T + (3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (2.03 - 0.597i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (-2.44 + 2.12i)T + (1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.746 - 2.54i)T + (-10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (0.439 - 3.05i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-5.09 + 0.732i)T + (18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (3.10 + 0.446i)T + (27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-7.94 - 5.10i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-4.16 - 1.90i)T + (24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (-2.96 - 6.49i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-6.68 - 10.4i)T + (-17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + 9.79T + 47T^{2} \) |
| 53 | \( 1 + (-0.793 - 2.70i)T + (-44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (-3.65 + 12.4i)T + (-49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (4.26 - 6.63i)T + (-25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (2.86 + 2.48i)T + (9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (-4.09 + 4.72i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (1.30 + 9.08i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (-1.58 - 0.465i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (3.67 + 1.68i)T + (54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-2.05 + 1.31i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-5.20 - 11.3i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−11.12310447825526050675030706461, −9.740057654248346601036147188837, −9.369970709631012532740056306183, −8.204370869464639345127153653111, −7.44461096143806783646628858572, −6.35474861041141352525912907675, −6.03538749036573786682085998553, −4.71416790598703042042730600082, −3.40122667878869398070520939095, −1.35017467669056973662573468031,
0.56291682590504221790035330841, 2.50100124271209065625982250142, 3.69726278597623556525909922185, 4.50350233552871795111644825470, 5.73603328759149073402012356229, 7.04191361870537196160099711687, 8.000641477236992615358683301185, 9.157297278735964703959874686159, 9.857263247939688969075370533014, 10.26779951662678370077298567787
