| L(s) = 1 | + (0.415 + 0.909i)2-s + (−0.959 + 0.281i)3-s + (−0.654 + 0.755i)4-s + (2.75 + 1.77i)5-s + (−0.654 − 0.755i)6-s + (−0.142 − 0.989i)7-s + (−0.959 − 0.281i)8-s + (0.841 − 0.540i)9-s + (−0.466 + 3.24i)10-s + (1.55 − 3.41i)11-s + (0.415 − 0.909i)12-s + (−0.870 + 6.05i)13-s + (0.841 − 0.540i)14-s + (−3.14 − 0.923i)15-s + (−0.142 − 0.989i)16-s + (3.15 + 3.64i)17-s + ⋯ |
| L(s) = 1 | + (0.293 + 0.643i)2-s + (−0.553 + 0.162i)3-s + (−0.327 + 0.377i)4-s + (1.23 + 0.792i)5-s + (−0.267 − 0.308i)6-s + (−0.0537 − 0.374i)7-s + (−0.339 − 0.0996i)8-s + (0.280 − 0.180i)9-s + (−0.147 + 1.02i)10-s + (0.469 − 1.02i)11-s + (0.119 − 0.262i)12-s + (−0.241 + 1.67i)13-s + (0.224 − 0.144i)14-s + (−0.811 − 0.238i)15-s + (−0.0355 − 0.247i)16-s + (0.765 + 0.883i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.357 - 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.357 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.04487 + 1.51857i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04487 + 1.51857i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.415 - 0.909i)T \) |
| 3 | \( 1 + (0.959 - 0.281i)T \) |
| 7 | \( 1 + (0.142 + 0.989i)T \) |
| 23 | \( 1 + (-3.00 - 3.74i)T \) |
| good | 5 | \( 1 + (-2.75 - 1.77i)T + (2.07 + 4.54i)T^{2} \) |
| 11 | \( 1 + (-1.55 + 3.41i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (0.870 - 6.05i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (-3.15 - 3.64i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (-1.00 + 1.15i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (4.86 + 5.61i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (-6.68 - 1.96i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (1.78 - 1.14i)T + (15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (-3.95 - 2.54i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (4.61 - 1.35i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 + (-0.865 - 6.01i)T + (-50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (-0.463 + 3.22i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (0.218 + 0.0640i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (-1.75 - 3.84i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (-1.93 - 4.23i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-1.30 + 1.51i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (-0.963 + 6.70i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (6.51 - 4.18i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (13.7 - 4.03i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (-1.00 - 0.643i)T + (40.2 + 88.2i)T^{2} \) |
| 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
−10.04703401122462141447965847798, −9.619167977762000740075887513338, −8.663481504136569884333105675368, −7.45907742199018952623727265551, −6.49460081083174145750693322900, −6.23049825009904911583960057445, −5.29565957602692526643257224927, −4.17108238660876337279754480031, −3.12640306884597130855746380576, −1.56786698763148022150934452324,
0.916125043261894791228174602638, 2.02724407100119795869044060064, 3.18293121563638567740226285713, 4.81944442578141559278064703462, 5.24025423423362795347826045660, 5.98831253445196448739114869829, 7.07813180592978861410480190933, 8.241052359067953184493672854762, 9.292282104045199528164670126199, 9.881805203196079472530296274166
