| L(s) = 1 | + (0.893 − 0.448i)2-s + (−0.530 − 1.64i)3-s + (0.597 − 0.802i)4-s + (−0.536 + 1.79i)5-s + (−1.21 − 1.23i)6-s + (1.99 − 4.62i)7-s + (0.173 − 0.984i)8-s + (−2.43 + 1.74i)9-s + (0.325 + 1.84i)10-s + (1.04 + 0.247i)11-s + (−1.63 − 0.559i)12-s + (−2.13 + 1.40i)13-s + (−0.292 − 5.02i)14-s + (3.24 − 0.0656i)15-s + (−0.286 − 0.957i)16-s + (5.49 + 1.99i)17-s + ⋯ |
| L(s) = 1 | + (0.631 − 0.317i)2-s + (−0.306 − 0.951i)3-s + (0.298 − 0.401i)4-s + (−0.240 + 0.801i)5-s + (−0.495 − 0.504i)6-s + (0.754 − 1.74i)7-s + (0.0613 − 0.348i)8-s + (−0.812 + 0.582i)9-s + (0.102 + 0.582i)10-s + (0.314 + 0.0745i)11-s + (−0.473 − 0.161i)12-s + (−0.590 + 0.388i)13-s + (−0.0782 − 1.34i)14-s + (0.836 − 0.0169i)15-s + (−0.0717 − 0.239i)16-s + (1.33 + 0.484i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.176 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.176 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.10868 - 0.927106i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.10868 - 0.927106i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.893 + 0.448i)T \) |
| 3 | \( 1 + (0.530 + 1.64i)T \) |
| good | 5 | \( 1 + (0.536 - 1.79i)T + (-4.17 - 2.74i)T^{2} \) |
| 7 | \( 1 + (-1.99 + 4.62i)T + (-4.80 - 5.09i)T^{2} \) |
| 11 | \( 1 + (-1.04 - 0.247i)T + (9.82 + 4.93i)T^{2} \) |
| 13 | \( 1 + (2.13 - 1.40i)T + (5.14 - 11.9i)T^{2} \) |
| 17 | \( 1 + (-5.49 - 1.99i)T + (13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (5.64 - 2.05i)T + (14.5 - 12.2i)T^{2} \) |
| 23 | \( 1 + (-1.77 - 4.10i)T + (-15.7 + 16.7i)T^{2} \) |
| 29 | \( 1 + (0.0699 - 1.20i)T + (-28.8 - 3.36i)T^{2} \) |
| 31 | \( 1 + (-6.69 - 0.782i)T + (30.1 + 7.14i)T^{2} \) |
| 37 | \( 1 + (-1.39 + 1.16i)T + (6.42 - 36.4i)T^{2} \) |
| 41 | \( 1 + (5.78 + 2.90i)T + (24.4 + 32.8i)T^{2} \) |
| 43 | \( 1 + (1.11 - 1.18i)T + (-2.50 - 42.9i)T^{2} \) |
| 47 | \( 1 + (11.9 - 1.39i)T + (45.7 - 10.8i)T^{2} \) |
| 53 | \( 1 + (-2.50 - 4.34i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.74 - 0.649i)T + (52.7 - 26.4i)T^{2} \) |
| 61 | \( 1 + (2.62 + 3.52i)T + (-17.4 + 58.4i)T^{2} \) |
| 67 | \( 1 + (0.636 + 10.9i)T + (-66.5 + 7.77i)T^{2} \) |
| 71 | \( 1 + (1.51 + 8.61i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (1.22 - 6.97i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-5.34 + 2.68i)T + (47.1 - 63.3i)T^{2} \) |
| 83 | \( 1 + (8.01 - 4.02i)T + (49.5 - 66.5i)T^{2} \) |
| 89 | \( 1 + (1.24 - 7.06i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (2.10 + 7.03i)T + (-81.0 + 53.3i)T^{2} \) |
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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
−12.64990700573181048923075724318, −11.66542738057825649162530167899, −10.87214693921778144439844117212, −10.14868677022663565520067054078, −8.021369453052110153770824325172, −7.22943100814509615894426747216, −6.40268215463179604456291249714, −4.79984756835672291033431999316, −3.45112209456895968134870763054, −1.52197809693834819267696396075,
2.80942776203676705726141428476, 4.63648482627436215928752745940, 5.16895926886528677382314910629, 6.25605242101963163388545623385, 8.254547384668164784916791296536, 8.779907576995347007473715383360, 10.04069929551228959154474635468, 11.51330662111701826954731666935, 12.02320174966135586065888135111, 12.82776963265618647220513584449
