L(s) = 1 | + (−1.12 − 0.408i)2-s + (−0.439 − 0.369i)4-s + (0.766 − 0.642i)5-s + (1.09 − 1.90i)7-s + (1.53 + 2.66i)8-s + (−1.12 + 0.408i)10-s + (−1.41 − 2.44i)11-s + (0.708 − 4.01i)13-s + (−2.00 + 1.68i)14-s + (−0.437 − 2.48i)16-s + (0.359 + 0.130i)17-s + (−2.75 + 3.37i)19-s − 0.574·20-s + (0.586 + 3.32i)22-s + (2.27 + 1.91i)23-s + ⋯ |
L(s) = 1 | + (−0.793 − 0.288i)2-s + (−0.219 − 0.184i)4-s + (0.342 − 0.287i)5-s + (0.415 − 0.718i)7-s + (0.543 + 0.941i)8-s + (−0.354 + 0.129i)10-s + (−0.426 − 0.738i)11-s + (0.196 − 1.11i)13-s + (−0.536 + 0.450i)14-s + (−0.109 − 0.620i)16-s + (0.0872 + 0.0317i)17-s + (−0.632 + 0.774i)19-s − 0.128·20-s + (0.124 + 0.708i)22-s + (0.475 + 0.398i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 + 0.623i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.781 + 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.259043 - 0.740483i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.259043 - 0.740483i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.766 + 0.642i)T \) |
| 19 | \( 1 + (2.75 - 3.37i)T \) |
good | 2 | \( 1 + (1.12 + 0.408i)T + (1.53 + 1.28i)T^{2} \) |
| 7 | \( 1 + (-1.09 + 1.90i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.41 + 2.44i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.708 + 4.01i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.359 - 0.130i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (-2.27 - 1.91i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-8.05 + 2.93i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-2.34 + 4.06i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 + (0.544 + 3.08i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-1.53 + 1.29i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (10.2 - 3.72i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (5.72 + 4.80i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (12.1 + 4.43i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-4.57 - 3.84i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (0.986 - 0.358i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-5.99 + 5.03i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (2.44 + 13.8i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (2.33 + 13.2i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (2.77 - 4.81i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.13 + 6.46i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (13.5 + 4.94i)T + (74.3 + 62.3i)T^{2} \) |
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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
−10.03567402085022125531585084170, −8.993057643981072199619487609925, −8.179420431087920429799616291754, −7.78699429545210999328576766679, −6.30757470128597589465190669214, −5.38738372275826208530240587544, −4.55825896805483160250504287619, −3.18301518369421365648034116918, −1.68216627210905227192490287707, −0.52634198691151071176185594253,
1.62526432903522753053941966382, 2.90094342381719263795446938514, 4.40349514151568529916714546097, 5.11389337607574791784808302664, 6.65077700608070146206542451924, 6.97439403636433911917134201311, 8.291490490016891878402715862092, 8.705882722064544785483845100269, 9.536021392087122148900436689912, 10.28602220802764532841138618992