L(s) = 1 | + (−0.618 − 1.61i)3-s − 3.23i·5-s − 1.23i·7-s + (−2.23 + 2.00i)9-s − 5.23·11-s + 4.47·13-s + (−5.23 + 2.00i)15-s + 2.47i·17-s + 0.763i·19-s + (−2.00 + 0.763i)21-s − 2.47·23-s − 5.47·25-s + (4.61 + 2.38i)27-s − 4.76i·29-s − 5.23i·31-s + ⋯ |
L(s) = 1 | + (−0.356 − 0.934i)3-s − 1.44i·5-s − 0.467i·7-s + (−0.745 + 0.666i)9-s − 1.57·11-s + 1.24·13-s + (−1.35 + 0.516i)15-s + 0.599i·17-s + 0.175i·19-s + (−0.436 + 0.166i)21-s − 0.515·23-s − 1.09·25-s + (0.888 + 0.458i)27-s − 0.884i·29-s − 0.940i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.356i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.162521 - 0.880956i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.162521 - 0.880956i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.618 + 1.61i)T \) |
good | 5 | \( 1 + 3.23iT - 5T^{2} \) |
| 7 | \( 1 + 1.23iT - 7T^{2} \) |
| 11 | \( 1 + 5.23T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 - 2.47iT - 17T^{2} \) |
| 19 | \( 1 - 0.763iT - 19T^{2} \) |
| 23 | \( 1 + 2.47T + 23T^{2} \) |
| 29 | \( 1 + 4.76iT - 29T^{2} \) |
| 31 | \( 1 + 5.23iT - 31T^{2} \) |
| 37 | \( 1 + 8.47T + 37T^{2} \) |
| 41 | \( 1 + 6.47iT - 41T^{2} \) |
| 43 | \( 1 + 7.23iT - 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 3.23iT - 53T^{2} \) |
| 59 | \( 1 - 1.23T + 59T^{2} \) |
| 61 | \( 1 + 0.472T + 61T^{2} \) |
| 67 | \( 1 - 9.70iT - 67T^{2} \) |
| 71 | \( 1 - 15.4T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 0.291iT - 79T^{2} \) |
| 83 | \( 1 + 2.76T + 83T^{2} \) |
| 89 | \( 1 + 4iT - 89T^{2} \) |
| 97 | \( 1 - 0.472T + 97T^{2} \) |
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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
−10.98334888161003133564084213731, −10.19696241470054390601148857267, −8.702375178644099375265570125136, −8.218054529458063796990288542526, −7.31788822670226719981440106942, −5.90825067597875359038969952359, −5.30617010774444546975566291979, −3.98058799524650802644509326142, −2.04424415332624413801861504578, −0.61237961889001587365307450298,
2.69981003140452455874462258290, 3.49651607573513232125593208464, 5.01193824464423276237475718191, 5.92109346824573554556271882355, 6.87775482818256847204336463769, 8.095712444197151852803827793526, 9.124738664149047183908370144373, 10.28861848829324456883825607556, 10.70907726085722303005520571532, 11.35910484359166104117477000068