| L(s) = 1 | + (−209. − 121. i)2-s + (2.11e4 + 3.66e4i)4-s + (6.97e4 − 4.02e4i)5-s + (−3.76e5 + 6.51e5i)7-s − 6.26e6i·8-s − 1.95e7·10-s + (−1.49e7 − 8.64e6i)11-s + (−2.09e7 − 3.63e7i)13-s + (1.57e8 − 9.11e7i)14-s + (−4.12e8 + 7.14e8i)16-s + 4.35e8i·17-s + 9.97e8·19-s + (2.94e9 + 1.70e9i)20-s + (2.09e9 + 3.62e9i)22-s + (1.81e9 − 1.04e9i)23-s + ⋯ |
| L(s) = 1 | + (−1.63 − 0.946i)2-s + (1.28 + 2.23i)4-s + (0.892 − 0.515i)5-s + (−0.457 + 0.791i)7-s − 2.98i·8-s − 1.95·10-s + (−0.768 − 0.443i)11-s + (−0.334 − 0.579i)13-s + (1.49 − 0.864i)14-s + (−1.53 + 2.66i)16-s + 1.06i·17-s + 1.11·19-s + (2.30 + 1.32i)20-s + (0.839 + 1.45i)22-s + (0.532 − 0.307i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.100 + 0.994i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(0.8276919398\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8276919398\) |
| \(L(8)\) |
|
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 | \( 1 + (209. + 121. i)T + (8.19e3 + 1.41e4i)T^{2} \) |
| 5 | \( 1 + (-6.97e4 + 4.02e4i)T + (3.05e9 - 5.28e9i)T^{2} \) |
| 7 | \( 1 + (3.76e5 - 6.51e5i)T + (-3.39e11 - 5.87e11i)T^{2} \) |
| 11 | \( 1 + (1.49e7 + 8.64e6i)T + (1.89e14 + 3.28e14i)T^{2} \) |
| 13 | \( 1 + (2.09e7 + 3.63e7i)T + (-1.96e15 + 3.40e15i)T^{2} \) |
| 17 | \( 1 - 4.35e8iT - 1.68e17T^{2} \) |
| 19 | \( 1 - 9.97e8T + 7.99e17T^{2} \) |
| 23 | \( 1 + (-1.81e9 + 1.04e9i)T + (5.79e18 - 1.00e19i)T^{2} \) |
| 29 | \( 1 + (2.31e10 + 1.33e10i)T + (1.48e20 + 2.57e20i)T^{2} \) |
| 31 | \( 1 + (-1.04e10 - 1.80e10i)T + (-3.78e20 + 6.55e20i)T^{2} \) |
| 37 | \( 1 - 5.64e10T + 9.01e21T^{2} \) |
| 41 | \( 1 + (-1.74e11 + 1.00e11i)T + (1.89e22 - 3.28e22i)T^{2} \) |
| 43 | \( 1 + (-2.99e10 + 5.19e10i)T + (-3.69e22 - 6.39e22i)T^{2} \) |
| 47 | \( 1 + (-1.62e11 - 9.38e10i)T + (1.28e23 + 2.22e23i)T^{2} \) |
| 53 | \( 1 - 5.89e11iT - 1.37e24T^{2} \) |
| 59 | \( 1 + (-2.04e12 + 1.17e12i)T + (3.09e24 - 5.36e24i)T^{2} \) |
| 61 | \( 1 + (2.24e11 - 3.88e11i)T + (-4.93e24 - 8.55e24i)T^{2} \) |
| 67 | \( 1 + (3.74e12 + 6.49e12i)T + (-1.83e25 + 3.18e25i)T^{2} \) |
| 71 | \( 1 + 4.02e12iT - 8.27e25T^{2} \) |
| 73 | \( 1 - 1.50e13T + 1.22e26T^{2} \) |
| 79 | \( 1 + (-1.58e13 + 2.73e13i)T + (-1.84e26 - 3.19e26i)T^{2} \) |
| 83 | \( 1 + (8.14e12 + 4.69e12i)T + (3.68e26 + 6.37e26i)T^{2} \) |
| 89 | \( 1 + 5.29e13iT - 1.95e27T^{2} \) |
| 97 | \( 1 + (-3.97e12 + 6.87e12i)T + (-3.26e27 - 5.65e27i)T^{2} \) |
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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
−13.17408914635938952564524599827, −12.31941123557368922892175737417, −10.87579053166917917855774746381, −9.767992343488984155447565237912, −8.950296344102516013579588083643, −7.73538559983677565618639391300, −5.77905566508961570756227482275, −3.06938029580611552386947681145, −1.95586704562387199769422750398, −0.59355317440068171987023550136,
0.845110597565759465098088104554, 2.37685653140822671533450315622, 5.44054942389345045899776817715, 6.84310589771455407500688301306, 7.56566640307604592974976765289, 9.429171150475908231964135836536, 9.940429007530341856833900640093, 11.15606274152806325794135471803, 13.58581828877874040952160931336, 14.70583039130873259369596499334
