L(s) = 1 | + (0.5 + 0.866i)2-s − 1.73i·3-s + (−0.499 + 0.866i)4-s + (1.5 − 2.59i)5-s + (1.49 − 0.866i)6-s + (−0.5 − 0.866i)7-s − 0.999·8-s − 2.99·9-s + 3·10-s + (3 + 5.19i)11-s + (1.49 + 0.866i)12-s + (−1 + 1.73i)13-s + (0.499 − 0.866i)14-s + (−4.5 − 2.59i)15-s + (−0.5 − 0.866i)16-s + 6·17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s − 0.999i·3-s + (−0.249 + 0.433i)4-s + (0.670 − 1.16i)5-s + (0.612 − 0.353i)6-s + (−0.188 − 0.327i)7-s − 0.353·8-s − 0.999·9-s + 0.948·10-s + (0.904 + 1.56i)11-s + (0.433 + 0.249i)12-s + (−0.277 + 0.480i)13-s + (0.133 − 0.231i)14-s + (−1.16 − 0.670i)15-s + (−0.125 − 0.216i)16-s + 1.45·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.28028 - 0.225747i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.28028 - 0.225747i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + 1.73iT \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
good | 5 | \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 7T + 19T^{2} \) |
| 23 | \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1 + 1.73i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.5 + 4.33i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 3T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + (2.5 + 4.33i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−13.20986910823196139391488959493, −12.54574775076249188563250726737, −11.88455747256029342077452353981, −9.865036300491147798837126901256, −8.943665352280116993837185077328, −7.72947248319515276352626506245, −6.71548736137882161307211737010, −5.61153587545137261138159368982, −4.28245679157121827719439111253, −1.77013939322723170388853964880,
2.78406084968574424703432640976, 3.75638913511956709659578632254, 5.58530102299595835378930166780, 6.34169381378257625831735145687, 8.475795705608970126586706196731, 9.578776388011096610624079001792, 10.52514854254777462354520007841, 11.06723454414901334002775045985, 12.24928004782496096231071561576, 13.65357224935547719695164869503