| L(s) = 1 | + (−1.94 − 0.454i)2-s + (−0.0785 − 2.99i)3-s + (3.58 + 1.76i)4-s + (0.792 − 0.288i)5-s + (−1.20 + 5.87i)6-s + (−1.04 − 5.93i)7-s + (−6.18 − 5.07i)8-s + (−8.98 + 0.471i)9-s + (−1.67 + 0.201i)10-s + (2.38 + 0.866i)11-s + (5.02 − 10.8i)12-s + (4.73 − 5.64i)13-s + (−0.657 + 12.0i)14-s + (−0.926 − 2.35i)15-s + (9.73 + 12.6i)16-s + (−16.4 − 9.48i)17-s + ⋯ |
| L(s) = 1 | + (−0.973 − 0.227i)2-s + (−0.0261 − 0.999i)3-s + (0.896 + 0.442i)4-s + (0.158 − 0.0576i)5-s + (−0.201 + 0.979i)6-s + (−0.149 − 0.847i)7-s + (−0.772 − 0.634i)8-s + (−0.998 + 0.0523i)9-s + (−0.167 + 0.0201i)10-s + (0.216 + 0.0787i)11-s + (0.418 − 0.908i)12-s + (0.364 − 0.433i)13-s + (−0.0469 + 0.859i)14-s + (−0.0617 − 0.156i)15-s + (0.608 + 0.793i)16-s + (−0.965 − 0.557i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.146i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.989 + 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0484310 - 0.655993i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0484310 - 0.655993i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.94 + 0.454i)T \) |
| 3 | \( 1 + (0.0785 + 2.99i)T \) |
| good | 5 | \( 1 + (-0.792 + 0.288i)T + (19.1 - 16.0i)T^{2} \) |
| 7 | \( 1 + (1.04 + 5.93i)T + (-46.0 + 16.7i)T^{2} \) |
| 11 | \( 1 + (-2.38 - 0.866i)T + (92.6 + 77.7i)T^{2} \) |
| 13 | \( 1 + (-4.73 + 5.64i)T + (-29.3 - 166. i)T^{2} \) |
| 17 | \( 1 + (16.4 + 9.48i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-27.5 + 15.9i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (39.0 + 6.89i)T + (497. + 180. i)T^{2} \) |
| 29 | \( 1 + (26.6 - 22.3i)T + (146. - 828. i)T^{2} \) |
| 31 | \( 1 + (2.03 - 11.5i)T + (-903. - 328. i)T^{2} \) |
| 37 | \( 1 + (52.8 + 30.5i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-20.7 + 24.6i)T + (-291. - 1.65e3i)T^{2} \) |
| 43 | \( 1 + (1.18 - 3.26i)T + (-1.41e3 - 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-27.3 + 4.81i)T + (2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 + 8.35T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-47.5 + 17.3i)T + (2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-20.9 + 3.68i)T + (3.49e3 - 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-44.6 + 53.2i)T + (-779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (14.8 + 8.55i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-0.682 - 1.18i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (21.8 - 18.3i)T + (1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-14.7 + 12.3i)T + (1.19e3 - 6.78e3i)T^{2} \) |
| 89 | \( 1 + (-90.3 + 52.1i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (86.4 + 31.4i)T + (7.20e3 + 6.04e3i)T^{2} \) |
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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
−11.51509327087752520720199078152, −10.74142937895072134181503679926, −9.566285635510725244698425767278, −8.644289439240077342266346371620, −7.45804527325176124599949692006, −6.98533170011683159724422139435, −5.68902115472267950679184774795, −3.50552859434380529779489007325, −1.95516275521190145134177277260, −0.47630054782563961319921696503,
2.16680641820427195259841450452, 3.81062099211590629866015819789, 5.59399948547914380056796512236, 6.24010556342456729030184570711, 7.891236157201205200365440135782, 8.797040392678648239795083311627, 9.620272845962679274179666273429, 10.27561604194697988410463949334, 11.51726443792352991270397467631, 11.95204791964231315822862896473
