L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (1.87 − 1.07i)5-s + (2.63 + 0.219i)7-s − 0.999·8-s + (1.87 + 1.07i)10-s + (2.19 + 2.48i)11-s − 3.53i·13-s + (1.12 + 2.39i)14-s + (−0.5 − 0.866i)16-s + (3.82 − 6.63i)17-s + (1.66 − 0.963i)19-s + 2.15i·20-s + (−1.05 + 3.14i)22-s + (−7.21 + 4.16i)23-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.836 − 0.482i)5-s + (0.996 + 0.0828i)7-s − 0.353·8-s + (0.591 + 0.341i)10-s + (0.661 + 0.750i)11-s − 0.981i·13-s + (0.301 + 0.639i)14-s + (−0.125 − 0.216i)16-s + (0.928 − 1.60i)17-s + (0.383 − 0.221i)19-s + 0.482i·20-s + (−0.225 + 0.670i)22-s + (−1.50 + 0.869i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.885 - 0.465i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.885 - 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.670813971\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.670813971\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.63 - 0.219i)T \) |
| 11 | \( 1 + (-2.19 - 2.48i)T \) |
good | 5 | \( 1 + (-1.87 + 1.07i)T + (2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 + 3.53iT - 13T^{2} \) |
| 17 | \( 1 + (-3.82 + 6.63i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.66 + 0.963i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (7.21 - 4.16i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 4.79T + 29T^{2} \) |
| 31 | \( 1 + (-2.97 + 5.14i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.55 + 2.69i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 9.00T + 41T^{2} \) |
| 43 | \( 1 - 2.48iT - 43T^{2} \) |
| 47 | \( 1 + (7.34 - 4.23i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.83 - 2.79i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.95 - 2.28i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-9.71 + 5.60i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.46 + 2.53i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 14.9iT - 71T^{2} \) |
| 73 | \( 1 + (-0.486 - 0.280i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.63 + 3.82i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 8.45T + 83T^{2} \) |
| 89 | \( 1 + (-0.609 + 0.351i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 1.23T + 97T^{2} \) |
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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
−9.787869381878754800674508096347, −8.725873058559035099972001852381, −7.87651177820464910497315039060, −7.31757095490750236483667205449, −6.19702478251663773605937219787, −5.29590098248133058485538921160, −4.96779039271344660993163974073, −3.77576747278863551666911669635, −2.44656980103047325162252302158, −1.19245310227349348403163820053,
1.40481603984092734068990082038, 2.10656957940247500377377994444, 3.45541574054921704770589293203, 4.25610175444060353776413364469, 5.32298184506283228915845362495, 6.16983233160206223711816933527, 6.75288619401460306906738331602, 8.320395376196877320283022725659, 8.514084198615524590150515201237, 9.951853119567619824025872065816