L(s) = 1 | + (−1.36 − 2.36i)2-s + (−0.5 + 0.866i)3-s + (−2.73 + 4.73i)4-s + 2.73·6-s + (−0.866 + 2.5i)7-s + 9.46·8-s + (−0.499 − 0.866i)9-s + (−0.366 + 0.633i)11-s + (−2.73 − 4.73i)12-s − 2.26·13-s + (7.09 − 1.36i)14-s + (−7.46 − 12.9i)16-s + (1.63 − 2.83i)17-s + (−1.36 + 2.36i)18-s + (−2.23 − 3.86i)19-s + ⋯ |
L(s) = 1 | + (−0.965 − 1.67i)2-s + (−0.288 + 0.499i)3-s + (−1.36 + 2.36i)4-s + 1.11·6-s + (−0.327 + 0.944i)7-s + 3.34·8-s + (−0.166 − 0.288i)9-s + (−0.110 + 0.191i)11-s + (−0.788 − 1.36i)12-s − 0.629·13-s + (1.89 − 0.365i)14-s + (−1.86 − 3.23i)16-s + (0.396 − 0.686i)17-s + (−0.321 + 0.557i)18-s + (−0.512 − 0.886i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.922 - 0.386i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.922 - 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0359668 + 0.178877i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0359668 + 0.178877i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.866 - 2.5i)T \) |
good | 2 | \( 1 + (1.36 + 2.36i)T + (-1 + 1.73i)T^{2} \) |
| 11 | \( 1 + (0.366 - 0.633i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2.26T + 13T^{2} \) |
| 17 | \( 1 + (-1.63 + 2.83i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.23 + 3.86i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.36 + 4.09i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 4.19T + 29T^{2} \) |
| 31 | \( 1 + (-0.232 + 0.401i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.59 + 2.76i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 0.732T + 41T^{2} \) |
| 43 | \( 1 + 3.19T + 43T^{2} \) |
| 47 | \( 1 + (-1 - 1.73i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.19 + 10.7i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.0980 + 0.169i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2 + 3.46i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7.33 - 12.6i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 6.19T + 71T^{2} \) |
| 73 | \( 1 + (-6.33 + 10.9i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.69 - 6.40i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 15.1T + 83T^{2} \) |
| 89 | \( 1 + (7.56 + 13.0i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
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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
−10.28594571085522490486337334775, −9.647715086344724749135802758872, −8.998476057543805380668524343476, −8.197925406214220725868553583055, −6.97672055338985661064281058813, −5.29895431281522139939526090243, −4.26160705236712672132013570803, −3.02780322225673365980808133131, −2.16602590777935336527781732797, −0.15905344153980337818924475531,
1.43164456206703976689329821328, 4.04885955873784853450992779165, 5.35336596357687700716879020563, 6.12767522862625567105627362265, 6.98123974872391439139782675325, 7.68703974669635499769986208650, 8.303779396959485159947060209854, 9.476314684838113692070540040026, 10.18801416546287949801815846011, 10.87681671641877793249693891816