L(s) = 1 | − i·3-s − 4i·5-s + 2·7-s − 9-s − 4i·11-s − 2i·13-s − 4·15-s − 2·17-s + 8i·19-s − 2i·21-s + 4·23-s − 11·25-s + i·27-s + 6·31-s − 4·33-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 1.78i·5-s + 0.755·7-s − 0.333·9-s − 1.20i·11-s − 0.554i·13-s − 1.03·15-s − 0.485·17-s + 1.83i·19-s − 0.436i·21-s + 0.834·23-s − 2.20·25-s + 0.192i·27-s + 1.07·31-s − 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.567863 - 1.37094i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.567863 - 1.37094i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
good | 5 | \( 1 + 4iT - 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 8iT - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 + 14iT - 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−9.896673563496253942721087298003, −8.821521414159894913227301420852, −8.278326865716987156084613925871, −7.83814253456007657478137667272, −6.30042210312539327070286376552, −5.46098441734282868973399766682, −4.75363724414827720741509638280, −3.50353875069208416710026970920, −1.76585453833573443385017399365, −0.77858141589193121347441626816,
2.16416084766877351097921162649, 3.03471852115787710647152141334, 4.32986925465745354573808228271, 5.06817891142303539060681617542, 6.66507801682956302120994236992, 6.89746921824129312899974225309, 7.976423808355417881524759290288, 9.135333050200086696728605841255, 9.916600997518144663362488225329, 10.70206185831713788930245676267