L(s) = 1 | + (−1.61 − 0.618i)3-s − 0.874·5-s − 1.41i·7-s + (2.23 + 2.00i)9-s + 5.11·11-s − 3.23·13-s + (1.41 + 0.540i)15-s − 4.91·17-s − 8.27i·19-s + (−0.874 + 2.28i)21-s + (−4.24 − 2.23i)23-s − 4.23·25-s + (−2.38 − 4.61i)27-s + 5.23i·29-s − 6·31-s + ⋯ |
L(s) = 1 | + (−0.934 − 0.356i)3-s − 0.390·5-s − 0.534i·7-s + (0.745 + 0.666i)9-s + 1.54·11-s − 0.897·13-s + (0.365 + 0.139i)15-s − 1.19·17-s − 1.89i·19-s + (−0.190 + 0.499i)21-s + (−0.884 − 0.466i)23-s − 0.847·25-s + (−0.458 − 0.888i)27-s + 0.972i·29-s − 1.07·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.751 + 0.660i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.751 + 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.197563 - 0.524169i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.197563 - 0.524169i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.61 + 0.618i)T \) |
| 23 | \( 1 + (4.24 + 2.23i)T \) |
good | 5 | \( 1 + 0.874T + 5T^{2} \) |
| 7 | \( 1 + 1.41iT - 7T^{2} \) |
| 11 | \( 1 - 5.11T + 11T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 + 4.91T + 17T^{2} \) |
| 19 | \( 1 + 8.27iT - 19T^{2} \) |
| 29 | \( 1 - 5.23iT - 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 10.7iT - 37T^{2} \) |
| 41 | \( 1 + 0.472iT - 41T^{2} \) |
| 43 | \( 1 + 10.0iT - 43T^{2} \) |
| 47 | \( 1 - 11.7iT - 47T^{2} \) |
| 53 | \( 1 - 0.874T + 53T^{2} \) |
| 59 | \( 1 + 8iT - 59T^{2} \) |
| 61 | \( 1 - 1.20iT - 61T^{2} \) |
| 67 | \( 1 - 6.53iT - 67T^{2} \) |
| 71 | \( 1 + 12.1iT - 71T^{2} \) |
| 73 | \( 1 + 7.23T + 73T^{2} \) |
| 79 | \( 1 - 6.65iT - 79T^{2} \) |
| 83 | \( 1 + 7.94T + 83T^{2} \) |
| 89 | \( 1 - 0.333T + 89T^{2} \) |
| 97 | \( 1 - 2.41iT - 97T^{2} \) |
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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
−10.79653691592681419685147513527, −9.559484211794862741469528414610, −8.792624759657184618209407196659, −7.23436270532500575385724476566, −7.05642359596689663178653356342, −5.94078079314602353153722979758, −4.68119015946928778833445014550, −3.98369827803966235198981914841, −2.05940171217713392051953755433, −0.35858432026172426610123530756,
1.76448485705016061065564578608, 3.74188980756035717074216153232, 4.43329276268403949784208836061, 5.73191377437448414482299424190, 6.37601774940551165236674752199, 7.42475414452967052396264675274, 8.556253077339284727752936124116, 9.620338448648443669825175153478, 10.10965937362194059206645575707, 11.44858685851881199808720967857