| L(s) = 1 | + (1.01 − 0.453i)2-s + (−2.79 + 0.594i)3-s + (−0.507 + 0.563i)4-s + (0.361 + 3.43i)5-s + (−2.57 + 1.87i)6-s + (−0.949 + 2.92i)8-s + (4.72 − 2.10i)9-s + (1.92 + 3.33i)10-s + (−1.73 − 2.82i)11-s + (1.08 − 1.87i)12-s + (−1.66 − 1.21i)13-s + (−3.05 − 9.39i)15-s + (0.199 + 1.89i)16-s + (−1.76 − 0.786i)17-s + (3.85 − 4.28i)18-s + (−1.08 − 1.20i)19-s + ⋯ |
| L(s) = 1 | + (0.719 − 0.320i)2-s + (−1.61 + 0.343i)3-s + (−0.253 + 0.281i)4-s + (0.161 + 1.53i)5-s + (−1.05 + 0.764i)6-s + (−0.335 + 1.03i)8-s + (1.57 − 0.701i)9-s + (0.609 + 1.05i)10-s + (−0.524 − 0.851i)11-s + (0.312 − 0.541i)12-s + (−0.462 − 0.335i)13-s + (−0.788 − 2.42i)15-s + (0.0499 + 0.474i)16-s + (−0.428 − 0.190i)17-s + (0.909 − 1.00i)18-s + (−0.249 − 0.276i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 + 0.300i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.953 + 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0494079 - 0.320841i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0494079 - 0.320841i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 + (1.73 + 2.82i)T \) |
| good | 2 | \( 1 + (-1.01 + 0.453i)T + (1.33 - 1.48i)T^{2} \) |
| 3 | \( 1 + (2.79 - 0.594i)T + (2.74 - 1.22i)T^{2} \) |
| 5 | \( 1 + (-0.361 - 3.43i)T + (-4.89 + 1.03i)T^{2} \) |
| 13 | \( 1 + (1.66 + 1.21i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (1.76 + 0.786i)T + (11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (1.08 + 1.20i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-0.403 + 0.698i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.46 - 7.58i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.0824 + 0.784i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (9.84 + 2.09i)T + (33.8 + 15.0i)T^{2} \) |
| 41 | \( 1 + (-0.657 + 2.02i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 3.08T + 43T^{2} \) |
| 47 | \( 1 + (5.06 + 5.62i)T + (-4.91 + 46.7i)T^{2} \) |
| 53 | \( 1 + (1.13 - 10.7i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (-2.20 + 2.44i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (0.112 + 1.07i)T + (-59.6 + 12.6i)T^{2} \) |
| 67 | \( 1 + (1.20 + 2.08i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.57 + 1.87i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (0.820 - 0.911i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (8.66 - 3.85i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (13.0 - 9.44i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (2.21 - 3.84i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.23 - 3.80i)T + (29.9 + 92.2i)T^{2} \) |
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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
−11.10977327303042294264489441193, −10.85121267710586015205939000290, −10.09253807008308676954643337465, −8.712047154566605310057180015188, −7.33182656421641648471661165709, −6.50533981116394531057649207700, −5.58990124679175486836332865825, −4.89805036922014061726639366486, −3.64213009111919534938681529370, −2.65995301775068568470725926686,
0.18355022610223586796021685656, 1.58555393474459590496160182785, 4.35398748234433252062670033493, 4.80961321006696522674608145267, 5.51145766466292468106066828225, 6.29752561800697958149808629885, 7.24926115919203659015608244693, 8.559111688546359133188508971049, 9.703987578125023536898525550603, 10.26856066946713006559616589456
