| L(s) = 1 | + (−1.07 + 2.11i)2-s + (1.72 + 0.128i)3-s + (−2.13 − 2.93i)4-s + (−0.134 + 0.847i)5-s + (−2.13 + 3.51i)6-s + (−2.44 + 2.86i)7-s + (3.80 − 0.603i)8-s + (2.96 + 0.445i)9-s + (−1.64 − 1.19i)10-s + (1.58 + 2.59i)11-s + (−3.30 − 5.34i)12-s + (0.0226 − 0.288i)13-s + (−3.41 − 8.25i)14-s + (−0.341 + 1.44i)15-s + (−0.585 + 1.80i)16-s + (1.61 − 6.74i)17-s + ⋯ |
| L(s) = 1 | + (−0.761 + 1.49i)2-s + (0.997 + 0.0744i)3-s + (−1.06 − 1.46i)4-s + (−0.0600 + 0.378i)5-s + (−0.870 + 1.43i)6-s + (−0.924 + 1.08i)7-s + (1.34 − 0.213i)8-s + (0.988 + 0.148i)9-s + (−0.520 − 0.378i)10-s + (0.479 + 0.781i)11-s + (−0.953 − 1.54i)12-s + (0.00629 − 0.0799i)13-s + (−0.913 − 2.20i)14-s + (−0.0880 + 0.373i)15-s + (−0.146 + 0.450i)16-s + (0.392 − 1.63i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.729 - 0.684i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.729 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.327430 + 0.827484i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.327430 + 0.827484i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.72 - 0.128i)T \) |
| 41 | \( 1 + (6.35 + 0.793i)T \) |
| good | 2 | \( 1 + (1.07 - 2.11i)T + (-1.17 - 1.61i)T^{2} \) |
| 5 | \( 1 + (0.134 - 0.847i)T + (-4.75 - 1.54i)T^{2} \) |
| 7 | \( 1 + (2.44 - 2.86i)T + (-1.09 - 6.91i)T^{2} \) |
| 11 | \( 1 + (-1.58 - 2.59i)T + (-4.99 + 9.80i)T^{2} \) |
| 13 | \( 1 + (-0.0226 + 0.288i)T + (-12.8 - 2.03i)T^{2} \) |
| 17 | \( 1 + (-1.61 + 6.74i)T + (-15.1 - 7.71i)T^{2} \) |
| 19 | \( 1 + (0.263 + 3.34i)T + (-18.7 + 2.97i)T^{2} \) |
| 23 | \( 1 + (-1.00 - 3.10i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (0.519 + 2.16i)T + (-25.8 + 13.1i)T^{2} \) |
| 31 | \( 1 + (-5.21 + 7.17i)T + (-9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (5.49 - 3.99i)T + (11.4 - 35.1i)T^{2} \) |
| 43 | \( 1 + (1.78 + 0.907i)T + (25.2 + 34.7i)T^{2} \) |
| 47 | \( 1 + (8.66 - 7.39i)T + (7.35 - 46.4i)T^{2} \) |
| 53 | \( 1 + (-9.81 + 2.35i)T + (47.2 - 24.0i)T^{2} \) |
| 59 | \( 1 + (-5.08 + 1.65i)T + (47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (4.78 + 9.38i)T + (-35.8 + 49.3i)T^{2} \) |
| 67 | \( 1 + (-0.950 + 1.55i)T + (-30.4 - 59.6i)T^{2} \) |
| 71 | \( 1 + (-1.79 + 1.10i)T + (32.2 - 63.2i)T^{2} \) |
| 73 | \( 1 + (3.68 - 3.68i)T - 73iT^{2} \) |
| 79 | \( 1 + (2.80 - 6.76i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + 0.612iT - 83T^{2} \) |
| 89 | \( 1 + (9.46 + 8.07i)T + (13.9 + 87.9i)T^{2} \) |
| 97 | \( 1 + (-5.27 - 3.23i)T + (44.0 + 86.4i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.21387544548702950143088091798, −13.22189588275293460017821459166, −11.85611314957171503808792107338, −9.774577987289465359266856414449, −9.465266433061662504930441276803, −8.478050894207496169161712392998, −7.26909411706799626659910437398, −6.59037594283159126291155380207, −5.04875896337355392577961270177, −2.90311675667836752524753215745,
1.32042204637125664356913141052, 3.24539133337117886257606045004, 3.97714048620233397662082689265, 6.72198963925762383732278611879, 8.324424426962583424480419302213, 8.837126657021098261212116609325, 10.23909560304901907804781089118, 10.41673379189270158945762783175, 12.11580187589768767981739814563, 12.85965254147318416634507416881
