| L(s) = 1 | + (−1.55 + 0.694i)2-s + (−2.24 + 0.476i)3-s + (0.611 − 0.679i)4-s + (−0.425 − 4.04i)5-s + (3.16 − 2.29i)6-s + (0.572 − 1.76i)8-s + (2.05 − 0.915i)9-s + (3.47 + 6.01i)10-s + (3.11 − 1.12i)11-s + (−1.04 + 1.81i)12-s + (−2.65 − 1.92i)13-s + (2.87 + 8.86i)15-s + (0.521 + 4.96i)16-s + (−1.20 − 0.537i)17-s + (−2.57 + 2.85i)18-s + (1.44 + 1.60i)19-s + ⋯ |
| L(s) = 1 | + (−1.10 + 0.490i)2-s + (−1.29 + 0.275i)3-s + (0.305 − 0.339i)4-s + (−0.190 − 1.80i)5-s + (1.29 − 0.938i)6-s + (0.202 − 0.623i)8-s + (0.685 − 0.305i)9-s + (1.09 + 1.90i)10-s + (0.940 − 0.340i)11-s + (−0.302 + 0.523i)12-s + (−0.735 − 0.534i)13-s + (0.743 + 2.28i)15-s + (0.130 + 1.24i)16-s + (−0.292 − 0.130i)17-s + (−0.606 + 0.673i)18-s + (0.331 + 0.368i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 + 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0174717 - 0.142373i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0174717 - 0.142373i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 + (-3.11 + 1.12i)T \) |
| good | 2 | \( 1 + (1.55 - 0.694i)T + (1.33 - 1.48i)T^{2} \) |
| 3 | \( 1 + (2.24 - 0.476i)T + (2.74 - 1.22i)T^{2} \) |
| 5 | \( 1 + (0.425 + 4.04i)T + (-4.89 + 1.03i)T^{2} \) |
| 13 | \( 1 + (2.65 + 1.92i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (1.20 + 0.537i)T + (11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (-1.44 - 1.60i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-0.933 + 1.61i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.0754 + 0.232i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.718 + 6.83i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (0.250 + 0.0531i)T + (33.8 + 15.0i)T^{2} \) |
| 41 | \( 1 + (1.77 - 5.45i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 8.01T + 43T^{2} \) |
| 47 | \( 1 + (2.72 + 3.02i)T + (-4.91 + 46.7i)T^{2} \) |
| 53 | \( 1 + (-0.523 + 4.97i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (0.658 - 0.730i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (0.193 + 1.83i)T + (-59.6 + 12.6i)T^{2} \) |
| 67 | \( 1 + (-1.50 - 2.60i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (5.23 - 3.80i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (6.45 - 7.17i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (5.05 - 2.25i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (1.67 - 1.21i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (8.31 - 14.4i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.09 - 1.51i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.05662888622108094166084753056, −9.536402607726581711245999691292, −8.607260013192480380601241891955, −8.006744530443736286076741169037, −6.79919566337843488823897407013, −5.78648998713689109126379781768, −4.91843003316714902290330028879, −4.05381147156220897241801999231, −1.18834896460022652951161534047, −0.16736872034262737328587719625,
1.71986921706282225105096591071, 3.09839007628684410904633983042, 4.73259149703252156345648859581, 6.01255423382809012639540607312, 6.94932692171982888153013593699, 7.31129505828155940989548604526, 8.784977285251582994086064299185, 9.820604295569897941473350670647, 10.43116228630424335210921856407, 11.18034025670370013741581283668
