L(s) = 1 | − 1.41·2-s + 2.00·4-s + 2.23i·5-s + 1.47i·7-s − 2.82·8-s − 3.16i·10-s − 6.04i·11-s − 5.21·13-s − 2.08i·14-s + 4.00·16-s + 15.7i·17-s − 4.82i·19-s + 4.47i·20-s + 8.55i·22-s + (2.58 − 22.8i)23-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.500·4-s + 0.447i·5-s + 0.210i·7-s − 0.353·8-s − 0.316i·10-s − 0.549i·11-s − 0.401·13-s − 0.149i·14-s + 0.250·16-s + 0.923i·17-s − 0.254i·19-s + 0.223i·20-s + 0.388i·22-s + (0.112 − 0.993i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.112i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.993 + 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.230973214\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.230973214\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| 23 | \( 1 + (-2.58 + 22.8i)T \) |
good | 7 | \( 1 - 1.47iT - 49T^{2} \) |
| 11 | \( 1 + 6.04iT - 121T^{2} \) |
| 13 | \( 1 + 5.21T + 169T^{2} \) |
| 17 | \( 1 - 15.7iT - 289T^{2} \) |
| 19 | \( 1 + 4.82iT - 361T^{2} \) |
| 29 | \( 1 - 23.4T + 841T^{2} \) |
| 31 | \( 1 + 20.4T + 961T^{2} \) |
| 37 | \( 1 + 15.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 20.3T + 1.68e3T^{2} \) |
| 43 | \( 1 - 38.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 13.8T + 2.20e3T^{2} \) |
| 53 | \( 1 + 38.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 33.5T + 3.48e3T^{2} \) |
| 61 | \( 1 + 100. iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 32.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 24.1T + 5.04e3T^{2} \) |
| 73 | \( 1 - 15.1T + 5.32e3T^{2} \) |
| 79 | \( 1 - 11.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 44.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 111. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 154. iT - 9.40e3T^{2} \) |
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
−8.796608067229708505837255139188, −8.304253722320617954219199904344, −7.44194451322226098194498154280, −6.62246693367901263176920678732, −6.00930230738205034215061919702, −4.99172718007601657271361008690, −3.82964912844089822594676059119, −2.85313542376066918274745499191, −1.95478019515995427346737876737, −0.58384213304114387610903901916,
0.72374894628099376715807645338, 1.82616159120051705518405608794, 2.88818506258979705552866325450, 4.03571060421203072745987525896, 5.01565958019287447046795882546, 5.76870102345238365478551523984, 6.95937188638411145850691434932, 7.36425560014234294083571430468, 8.233876265716833184643261488581, 9.034918534752171980229648827838