| L(s) = 1 | + (1.08 + 0.912i)2-s + (0.906 + 1.18i)3-s + (0.334 + 1.97i)4-s + (1.00 − 1.31i)5-s + (−0.0986 + 2.10i)6-s + (−2.23 + 1.41i)7-s + (−1.43 + 2.43i)8-s + (0.203 − 0.758i)9-s + (2.28 − 0.498i)10-s + (0.146 − 1.10i)11-s + (−2.02 + 2.18i)12-s + (0.522 − 1.26i)13-s + (−3.70 − 0.511i)14-s + 2.45·15-s + (−3.77 + 1.31i)16-s + (1.92 − 3.34i)17-s + ⋯ |
| L(s) = 1 | + (0.763 + 0.645i)2-s + (0.523 + 0.681i)3-s + (0.167 + 0.985i)4-s + (0.449 − 0.585i)5-s + (−0.0402 + 0.858i)6-s + (−0.844 + 0.534i)7-s + (−0.508 + 0.861i)8-s + (0.0677 − 0.252i)9-s + (0.721 − 0.157i)10-s + (0.0440 − 0.334i)11-s + (−0.584 + 0.629i)12-s + (0.144 − 0.349i)13-s + (−0.990 − 0.136i)14-s + 0.634·15-s + (−0.944 + 0.329i)16-s + (0.468 − 0.810i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.132 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.132 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.53723 + 1.34595i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.53723 + 1.34595i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.08 - 0.912i)T \) |
| 7 | \( 1 + (2.23 - 1.41i)T \) |
| good | 3 | \( 1 + (-0.906 - 1.18i)T + (-0.776 + 2.89i)T^{2} \) |
| 5 | \( 1 + (-1.00 + 1.31i)T + (-1.29 - 4.82i)T^{2} \) |
| 11 | \( 1 + (-0.146 + 1.10i)T + (-10.6 - 2.84i)T^{2} \) |
| 13 | \( 1 + (-0.522 + 1.26i)T + (-9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 + (-1.92 + 3.34i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.0505 + 0.383i)T + (-18.3 + 4.91i)T^{2} \) |
| 23 | \( 1 + (3.87 + 1.03i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (0.846 - 2.04i)T + (-20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + (1.81 - 3.14i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.40 + 2.61i)T + (9.57 + 35.7i)T^{2} \) |
| 41 | \( 1 + (-3.82 + 3.82i)T - 41iT^{2} \) |
| 43 | \( 1 + (-1.98 + 0.821i)T + (30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (-8.69 + 5.01i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (11.4 + 1.50i)T + (51.1 + 13.7i)T^{2} \) |
| 59 | \( 1 + (1.42 - 10.8i)T + (-56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (-1.01 - 7.70i)T + (-58.9 + 15.7i)T^{2} \) |
| 67 | \( 1 + (10.8 - 8.34i)T + (17.3 - 64.7i)T^{2} \) |
| 71 | \( 1 + (-4.38 - 4.38i)T + 71iT^{2} \) |
| 73 | \( 1 + (3.29 + 12.2i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (5.09 + 8.82i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.46 - 0.607i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (3.10 - 11.5i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + 3.79iT - 97T^{2} \) |
| 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
−12.60796851284621503563854979293, −11.93984626921817523148001074440, −10.38270805031357974219964464207, −9.172809661394878139738821252924, −8.804059748826871613694619835268, −7.34233919176983358058321146075, −6.08251093673457500627671131401, −5.23704502911112417269588760287, −3.87561544982294510995105595027, −2.86436552072822479051199688154,
1.79821877196161393333959394011, 3.00877660421412630023028107793, 4.25983202571232470135388674902, 5.96061514407791955322307370688, 6.73908952707633306587248423170, 7.85385878932659460899466342160, 9.461884560217206662960136544159, 10.22973618482493832728927101039, 11.05126234439194525524542889432, 12.42555852266577934889837289029
