| L(s) = 1 | + (−1.11 − 0.870i)2-s + (1.97 − 0.392i)3-s + (0.485 + 1.94i)4-s + (−0.153 − 0.229i)5-s + (−2.53 − 1.27i)6-s + (−0.843 − 0.349i)7-s + (1.14 − 2.58i)8-s + (0.961 − 0.398i)9-s + (−0.0287 + 0.388i)10-s + (0.575 − 2.89i)11-s + (1.71 + 3.63i)12-s + (−3.63 + 5.43i)13-s + (0.636 + 1.12i)14-s + (−0.391 − 0.391i)15-s + (−3.52 + 1.88i)16-s + (−3.22 + 3.22i)17-s + ⋯ |
| L(s) = 1 | + (−0.788 − 0.615i)2-s + (1.13 − 0.226i)3-s + (0.242 + 0.970i)4-s + (−0.0684 − 0.102i)5-s + (−1.03 − 0.521i)6-s + (−0.318 − 0.131i)7-s + (0.405 − 0.914i)8-s + (0.320 − 0.132i)9-s + (−0.00907 + 0.122i)10-s + (0.173 − 0.872i)11-s + (0.496 + 1.04i)12-s + (−1.00 + 1.50i)13-s + (0.169 + 0.300i)14-s + (−0.101 − 0.101i)15-s + (−0.882 + 0.471i)16-s + (−0.783 + 0.783i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.740 + 0.671i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.740 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.759046 - 0.293015i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.759046 - 0.293015i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.11 + 0.870i)T \) |
| good | 3 | \( 1 + (-1.97 + 0.392i)T + (2.77 - 1.14i)T^{2} \) |
| 5 | \( 1 + (0.153 + 0.229i)T + (-1.91 + 4.61i)T^{2} \) |
| 7 | \( 1 + (0.843 + 0.349i)T + (4.94 + 4.94i)T^{2} \) |
| 11 | \( 1 + (-0.575 + 2.89i)T + (-10.1 - 4.20i)T^{2} \) |
| 13 | \( 1 + (3.63 - 5.43i)T + (-4.97 - 12.0i)T^{2} \) |
| 17 | \( 1 + (3.22 - 3.22i)T - 17iT^{2} \) |
| 19 | \( 1 + (-5.20 - 3.47i)T + (7.27 + 17.5i)T^{2} \) |
| 23 | \( 1 + (2.33 + 5.64i)T + (-16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (-0.693 - 3.48i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + 3.92iT - 31T^{2} \) |
| 37 | \( 1 + (-4.35 + 2.90i)T + (14.1 - 34.1i)T^{2} \) |
| 41 | \( 1 + (0.653 + 1.57i)T + (-28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (3.96 + 0.788i)T + (39.7 + 16.4i)T^{2} \) |
| 47 | \( 1 + (-3.42 + 3.42i)T - 47iT^{2} \) |
| 53 | \( 1 + (0.321 - 1.61i)T + (-48.9 - 20.2i)T^{2} \) |
| 59 | \( 1 + (-3.43 - 5.13i)T + (-22.5 + 54.5i)T^{2} \) |
| 61 | \( 1 + (-11.5 + 2.29i)T + (56.3 - 23.3i)T^{2} \) |
| 67 | \( 1 + (-6.90 + 1.37i)T + (61.8 - 25.6i)T^{2} \) |
| 71 | \( 1 + (-2.53 - 1.05i)T + (50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-2.05 + 0.851i)T + (51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-4.21 - 4.21i)T + 79iT^{2} \) |
| 83 | \( 1 + (8.64 + 5.77i)T + (31.7 + 76.6i)T^{2} \) |
| 89 | \( 1 + (3.87 - 9.35i)T + (-62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + 1.83iT - 97T^{2} \) |
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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
−14.55226802492106424509994079730, −13.73124813197547479708312051920, −12.51317488628745216690432485902, −11.41756636582642695698137219218, −9.978777024321619510761229041769, −8.946419882138517716032184592425, −8.131184975415683353647823100322, −6.78356889844300243762193042021, −3.89630350451076708339536159927, −2.32702659086190799155540776491,
2.79285320000749333645888422424, 5.18763654716117222055690415946, 7.11136658808952407187693460414, 7.995599436010275268047873362392, 9.390195526014697093877495918288, 9.840824967435764665631885910747, 11.51103634312942235039344555263, 13.21419857719191352157613897158, 14.33828993042357696952935996999, 15.29896070605685272057208299076
