| L(s) = 1 | − 4.19i·5-s − 11.8·7-s − 11.9i·11-s − 2.47·13-s − 13.8i·17-s − 3.77·19-s + 26.8i·23-s + 7.41·25-s − 41.2i·29-s − 34.8·31-s + 49.6i·35-s − 67.0·37-s − 5.08i·41-s − 46.3·43-s − 93.2i·47-s + ⋯ |
| L(s) = 1 | − 0.838i·5-s − 1.69·7-s − 1.08i·11-s − 0.190·13-s − 0.814i·17-s − 0.198·19-s + 1.16i·23-s + 0.296·25-s − 1.42i·29-s − 1.12·31-s + 1.41i·35-s − 1.81·37-s − 0.124i·41-s − 1.07·43-s − 1.98i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.05253246728\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05253246728\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 4.19iT - 25T^{2} \) |
| 7 | \( 1 + 11.8T + 49T^{2} \) |
| 11 | \( 1 + 11.9iT - 121T^{2} \) |
| 13 | \( 1 + 2.47T + 169T^{2} \) |
| 17 | \( 1 + 13.8iT - 289T^{2} \) |
| 19 | \( 1 + 3.77T + 361T^{2} \) |
| 23 | \( 1 - 26.8iT - 529T^{2} \) |
| 29 | \( 1 + 41.2iT - 841T^{2} \) |
| 31 | \( 1 + 34.8T + 961T^{2} \) |
| 37 | \( 1 + 67.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + 5.08iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 46.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + 93.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 51.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 14.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 51.2T + 3.72e3T^{2} \) |
| 67 | \( 1 - 13.5T + 4.48e3T^{2} \) |
| 71 | \( 1 + 32.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 40.6T + 5.32e3T^{2} \) |
| 79 | \( 1 - 74.2T + 6.24e3T^{2} \) |
| 83 | \( 1 - 153. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 20.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 47.7T + 9.40e3T^{2} \) |
| 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
−8.840097687733339828573524643374, −8.195479256067111836713932853964, −7.09348556664236613742104916810, −6.57896134207455658193998225312, −5.58057052916495974464506093239, −5.15284460967072891406450713330, −3.73590627032557400820873600510, −3.38373703869983329463180138397, −2.19317389536709385310059098265, −0.73207019467941412108805934721,
0.01628586095118190616331008347, 1.75020571296516566584300470401, 2.83544974217131550835329767903, 3.41672463479946016762377469313, 4.34955330932617030065823125941, 5.40717847817413368923922611329, 6.41404934377398260323463920349, 6.80703817221068808355597149768, 7.33908705614391846479875423402, 8.503389666627562397372934145817
