L(s) = 1 | + (1.03 − 0.967i)2-s + (0.128 − 1.99i)4-s + (1.25 − 2.16i)5-s + (−1.36 + 2.26i)7-s + (−1.79 − 2.18i)8-s + (−0.805 − 3.44i)10-s + (−2.83 − 4.91i)11-s + 5.31·13-s + (0.787 + 3.65i)14-s + (−3.96 − 0.512i)16-s + (0.393 − 0.227i)17-s + (−3.19 − 1.84i)19-s + (−4.16 − 2.77i)20-s + (−7.68 − 2.32i)22-s + (4.43 + 2.56i)23-s + ⋯ |
L(s) = 1 | + (0.729 − 0.684i)2-s + (0.0642 − 0.997i)4-s + (0.559 − 0.969i)5-s + (−0.515 + 0.857i)7-s + (−0.635 − 0.771i)8-s + (−0.254 − 1.08i)10-s + (−0.855 − 1.48i)11-s + 1.47·13-s + (0.210 + 0.977i)14-s + (−0.991 − 0.128i)16-s + (0.0955 − 0.0551i)17-s + (−0.733 − 0.423i)19-s + (−0.931 − 0.620i)20-s + (−1.63 − 0.495i)22-s + (0.924 + 0.533i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.481 + 0.876i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.04775 - 1.77170i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.04775 - 1.77170i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.03 + 0.967i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.36 - 2.26i)T \) |
good | 5 | \( 1 + (-1.25 + 2.16i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2.83 + 4.91i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 5.31T + 13T^{2} \) |
| 17 | \( 1 + (-0.393 + 0.227i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.19 + 1.84i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.43 - 2.56i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 2.57iT - 29T^{2} \) |
| 31 | \( 1 + (3.00 + 5.20i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-7.80 - 4.50i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 4.65iT - 41T^{2} \) |
| 43 | \( 1 - 3.66T + 43T^{2} \) |
| 47 | \( 1 + (0.478 - 0.829i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (5.41 - 3.12i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-8.76 + 5.06i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.50 + 4.33i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.65 - 8.05i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 7.35iT - 71T^{2} \) |
| 73 | \( 1 + (-5.93 + 3.42i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.71 - 4.45i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 1.96iT - 83T^{2} \) |
| 89 | \( 1 + (5.91 + 3.41i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 3.71iT - 97T^{2} \) |
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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
−11.03702929722915991895814706971, −9.715907825906665393087065601483, −8.981636992447223246258457447359, −8.316565444408763433391789373285, −6.38474947293369021587323045946, −5.75125452952876561401793646283, −5.06200901250755535328526285907, −3.62526309017378015666342258040, −2.60236527999040969419776516149, −1.03743484733497864938880000891,
2.34491598659545758488739436854, 3.53683862770110482387139612489, 4.52380861828157371832263231451, 5.81651705249047280724177460064, 6.66286277413492956565587191765, 7.23008481298840059290307684002, 8.240533689915239928712945573041, 9.485596838488724851366088044894, 10.55205046510302955237478994153, 10.96187308260191485602840554292