| 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
−12.82776963265618647220513584449, −12.02320174966135586065888135111, −11.51330662111701826954731666935, −10.04069929551228959154474635468, −8.779907576995347007473715383360, −8.254547384668164784916791296536, −6.25605242101963163388545623385, −5.16895926886528677382314910629, −4.63648482627436215928752745940, −2.80942776203676705726141428476,
1.52197809693834819267696396075, 3.45112209456895968134870763054, 4.79984756835672291033431999316, 6.40268215463179604456291249714, 7.22943100814509615894426747216, 8.021369453052110153770824325172, 10.14868677022663565520067054078, 10.87214693921778144439844117212, 11.66542738057825649162530167899, 12.64990700573181048923075724318
