| L(s) = 1 | + (−0.707 − 1.22i)2-s + (−1.5 − 0.866i)3-s + (−0.999 + 1.73i)4-s + (−1.93 + 1.11i)5-s + 2.44i·6-s + (−2.59 − 6.50i)7-s + 2.82·8-s + (1.5 + 2.59i)9-s + (2.73 + 1.58i)10-s + (−5.13 + 8.89i)11-s + (2.99 − 1.73i)12-s + 7.02i·13-s + (−6.12 + 7.77i)14-s + 3.87·15-s + (−2.00 − 3.46i)16-s + (27.4 + 15.8i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.5 − 0.288i)3-s + (−0.249 + 0.433i)4-s + (−0.387 + 0.223i)5-s + 0.408i·6-s + (−0.370 − 0.928i)7-s + 0.353·8-s + (0.166 + 0.288i)9-s + (0.273 + 0.158i)10-s + (−0.466 + 0.808i)11-s + (0.249 − 0.144i)12-s + 0.540i·13-s + (−0.437 + 0.555i)14-s + 0.258·15-s + (−0.125 − 0.216i)16-s + (1.61 + 0.933i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.514 - 0.857i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.514 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.482673 + 0.273143i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.482673 + 0.273143i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.707 + 1.22i)T \) |
| 3 | \( 1 + (1.5 + 0.866i)T \) |
| 5 | \( 1 + (1.93 - 1.11i)T \) |
| 7 | \( 1 + (2.59 + 6.50i)T \) |
| good | 11 | \( 1 + (5.13 - 8.89i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 7.02iT - 169T^{2} \) |
| 17 | \( 1 + (-27.4 - 15.8i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (26.9 - 15.5i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-11.8 - 20.5i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 - 9.19T + 841T^{2} \) |
| 31 | \( 1 + (-17.4 - 10.0i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (24.0 + 41.7i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 65.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 3.03T + 1.84e3T^{2} \) |
| 47 | \( 1 + (53.6 - 30.9i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (0.690 - 1.19i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (95.1 + 54.9i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (34.3 - 19.8i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (7.95 - 13.7i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 53.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-62.6 - 36.1i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (53.2 + 92.1i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 49.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (142. - 82.4i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 49.4iT - 9.40e3T^{2} \) |
| 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
−12.40348703514963019316484216064, −11.21142989617709864405699676828, −10.40074515415956715120255634534, −9.754353741546411753698943623034, −8.150543492157771577639308593846, −7.39310247173944391435917686841, −6.25330788601664294564362374545, −4.59619656899815189670024080450, −3.43955158430301019902229508629, −1.52505091249007077451594289916,
0.38804118052352755605831340381, 3.03384971086687030987615879955, 4.85055744307682108423320650745, 5.70505691271501065793498381362, 6.75412098304075279779525962124, 8.119969309923232925567529956397, 8.851066725745696517049173553783, 9.977884419525046384365150194963, 10.89591202595865789879475032390, 12.00470879121372298965996660784
