L(s) = 1 | + 1.41i·2-s − 2.00·4-s + 2i·5-s − 2.82i·8-s − 2.82·10-s + 7.07·13-s + 4.00·16-s − 2i·17-s − 4.00i·20-s + 25-s + 10.0i·26-s + 9.89i·29-s + 5.65i·32-s + 2.82·34-s + 12·37-s + ⋯ |
L(s) = 1 | + 0.999i·2-s − 1.00·4-s + 0.894i·5-s − 1.00i·8-s − 0.894·10-s + 1.96·13-s + 1.00·16-s − 0.485i·17-s − 0.894i·20-s + 0.200·25-s + 1.96i·26-s + 1.83i·29-s + 1.00i·32-s + 0.485·34-s + 1.97·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.643280974\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.643280974\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 2iT - 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 7.07T + 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 9.89iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 12T + 37T^{2} \) |
| 41 | \( 1 + 8iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 12.7iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 15.5T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 7.07T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 16iT - 89T^{2} \) |
| 97 | \( 1 - 18.3T + 97T^{2} \) |
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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
−9.155721806403398768800806838786, −8.857355401257650190569389882899, −7.80874456641121904175974254014, −7.17457899748758513362373622160, −6.32956978148163461379656360116, −5.87136029994254922647475097805, −4.75798195623369930991834351556, −3.76150245797625627700967242535, −2.98436293698379732212671002035, −1.17045453004939814686854578233,
0.78552115538202780571156344868, 1.68852368127596592519540870803, 2.99887407347835740493424906111, 4.02452422995861658736526581565, 4.57582245834994345713036511453, 5.72287064771039191265290200714, 6.31623916961709260011527901753, 7.991257448755049527530440139740, 8.304578622662185770140409891221, 9.144720655472036548419316685397