| L(s) = 1 | + 70.1i·2-s + 1.14e4·4-s + 9.60e4i·5-s − 2.29e5·7-s + 1.95e6i·8-s − 6.73e6·10-s + 1.06e7i·11-s − 3.57e7·13-s − 1.60e7i·14-s + 5.06e7·16-s − 1.55e8i·17-s + 5.06e8·19-s + 1.10e9i·20-s − 7.48e8·22-s + 5.97e9i·23-s + ⋯ |
| L(s) = 1 | + 0.548i·2-s + 0.699·4-s + 1.22i·5-s − 0.278·7-s + 0.931i·8-s − 0.673·10-s + 0.547i·11-s − 0.570·13-s − 0.152i·14-s + 0.188·16-s − 0.379i·17-s + 0.566·19-s + 0.859i·20-s − 0.300·22-s + 1.75i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(1.649923854\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.649923854\) |
| \(L(8)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 70.1iT - 1.63e4T^{2} \) |
| 5 | \( 1 - 9.60e4iT - 6.10e9T^{2} \) |
| 7 | \( 1 + 2.29e5T + 6.78e11T^{2} \) |
| 11 | \( 1 - 1.06e7iT - 3.79e14T^{2} \) |
| 13 | \( 1 + 3.57e7T + 3.93e15T^{2} \) |
| 17 | \( 1 + 1.55e8iT - 1.68e17T^{2} \) |
| 19 | \( 1 - 5.06e8T + 7.99e17T^{2} \) |
| 23 | \( 1 - 5.97e9iT - 1.15e19T^{2} \) |
| 29 | \( 1 + 1.99e10iT - 2.97e20T^{2} \) |
| 31 | \( 1 + 4.17e10T + 7.56e20T^{2} \) |
| 37 | \( 1 + 3.21e10T + 9.01e21T^{2} \) |
| 41 | \( 1 + 3.07e11iT - 3.79e22T^{2} \) |
| 43 | \( 1 + 4.40e11T + 7.38e22T^{2} \) |
| 47 | \( 1 + 5.56e10iT - 2.56e23T^{2} \) |
| 53 | \( 1 - 2.05e12iT - 1.37e24T^{2} \) |
| 59 | \( 1 - 1.68e12iT - 6.19e24T^{2} \) |
| 61 | \( 1 + 1.40e12T + 9.87e24T^{2} \) |
| 67 | \( 1 - 3.62e12T + 3.67e25T^{2} \) |
| 71 | \( 1 - 7.21e12iT - 8.27e25T^{2} \) |
| 73 | \( 1 - 1.62e13T + 1.22e26T^{2} \) |
| 79 | \( 1 - 1.84e13T + 3.68e26T^{2} \) |
| 83 | \( 1 + 2.17e13iT - 7.36e26T^{2} \) |
| 89 | \( 1 - 5.27e13iT - 1.95e27T^{2} \) |
| 97 | \( 1 + 1.06e14T + 6.52e27T^{2} \) |
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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
−14.94832684559542690694492809763, −13.82642662643173047311687071596, −12.00165370381648175085803198408, −10.94365085419759278779864886925, −9.674218244256886003108810796880, −7.57052957276382955578116273899, −6.90363132603266501546393967507, −5.52329553186400640106126602255, −3.29518960340929097363840867223, −2.03523267117628102918366145156,
0.45618296817073361982064137311, 1.72324537101499645735644323422, 3.31964935216139471836199231909, 5.02349429110505446761466380083, 6.64101169731220395374741062366, 8.314653869722091109380520577017, 9.673330935928662647405118253912, 11.01666139526510849966085045511, 12.31473753183630238276900544086, 12.97932168067799029842505377186
