| L(s) = 1 | + (−0.330 − 0.943i)2-s + (−0.00699 + 0.0199i)3-s + (−0.781 + 0.623i)4-s + (−1.87 + 1.21i)5-s + 0.0211·6-s + (−1.40 − 1.40i)7-s + (0.846 + 0.532i)8-s + (2.34 + 1.87i)9-s + (1.76 + 1.37i)10-s + (2.79 − 3.50i)11-s + (−0.00699 − 0.0199i)12-s + (2.13 − 3.39i)13-s + (−0.860 + 1.78i)14-s + (−0.0111 − 0.0460i)15-s + (0.222 − 0.974i)16-s + (−2.96 − 4.72i)17-s + ⋯ |
| L(s) = 1 | + (−0.233 − 0.667i)2-s + (−0.00403 + 0.0115i)3-s + (−0.390 + 0.311i)4-s + (−0.839 + 0.543i)5-s + 0.00864·6-s + (−0.530 − 0.530i)7-s + (0.299 + 0.188i)8-s + (0.781 + 0.623i)9-s + (0.558 + 0.433i)10-s + (0.843 − 1.05i)11-s + (−0.00201 − 0.00577i)12-s + (0.590 − 0.940i)13-s + (−0.230 + 0.477i)14-s + (−0.00287 − 0.0118i)15-s + (0.0556 − 0.243i)16-s + (−0.720 − 1.14i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.335 + 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.335 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.512649 - 0.726709i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.512649 - 0.726709i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.330 + 0.943i)T \) |
| 5 | \( 1 + (1.87 - 1.21i)T \) |
| 43 | \( 1 + (-0.688 + 6.52i)T \) |
| good | 3 | \( 1 + (0.00699 - 0.0199i)T + (-2.34 - 1.87i)T^{2} \) |
| 7 | \( 1 + (1.40 + 1.40i)T + 7iT^{2} \) |
| 11 | \( 1 + (-2.79 + 3.50i)T + (-2.44 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-2.13 + 3.39i)T + (-5.64 - 11.7i)T^{2} \) |
| 17 | \( 1 + (2.96 + 4.72i)T + (-7.37 + 15.3i)T^{2} \) |
| 19 | \( 1 + (3.98 + 4.99i)T + (-4.22 + 18.5i)T^{2} \) |
| 23 | \( 1 + (2.75 - 0.310i)T + (22.4 - 5.11i)T^{2} \) |
| 29 | \( 1 + (-6.29 - 3.02i)T + (18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (0.491 + 0.236i)T + (19.3 + 24.2i)T^{2} \) |
| 37 | \( 1 + (-1.86 - 1.86i)T + 37iT^{2} \) |
| 41 | \( 1 + (-4.55 - 2.19i)T + (25.5 + 32.0i)T^{2} \) |
| 47 | \( 1 + (-1.67 - 0.188i)T + (45.8 + 10.4i)T^{2} \) |
| 53 | \( 1 + (1.63 - 1.02i)T + (22.9 - 47.7i)T^{2} \) |
| 59 | \( 1 + (5.67 + 1.29i)T + (53.1 + 25.5i)T^{2} \) |
| 61 | \( 1 + (-2.22 - 4.62i)T + (-38.0 + 47.6i)T^{2} \) |
| 67 | \( 1 + (2.45 + 0.276i)T + (65.3 + 14.9i)T^{2} \) |
| 71 | \( 1 + (-6.21 + 4.95i)T + (15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (9.28 + 5.83i)T + (31.6 + 65.7i)T^{2} \) |
| 79 | \( 1 + 0.739iT - 79T^{2} \) |
| 83 | \( 1 + (-11.8 - 4.13i)T + (64.8 + 51.7i)T^{2} \) |
| 89 | \( 1 + (13.8 - 6.67i)T + (55.4 - 69.5i)T^{2} \) |
| 97 | \( 1 + (0.866 + 7.68i)T + (-94.5 + 21.5i)T^{2} \) |
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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
−10.85958674365514416420076828986, −10.31197309262826849957852161839, −9.097536143261276597734935523796, −8.250945054865395888086709735306, −7.22449680125657322633514734542, −6.40620920277379984370834274823, −4.67232671779089676032561012481, −3.74623432207476167994352814866, −2.74203327269322449748253045447, −0.66248810204364495593271595589,
1.61326724324033898712956517408, 4.06164621383874903497018523186, 4.29784988575056698996432255683, 6.16671831900468497937733630367, 6.62803687625225650933692234421, 7.79951056087760394400241052872, 8.740228986973797099608687827771, 9.369307095664260980828622979850, 10.29890716229456737885126152454, 11.59449448662810671851548451501
