L(s) = 1 | + 2i·5-s − 2·7-s + 4i·11-s − 2i·13-s + 4·17-s + 4i·19-s + 8·23-s + 25-s − 6i·29-s − 6·31-s − 4i·35-s − 2i·37-s − 12·41-s + 12i·43-s − 8·47-s + ⋯ |
L(s) = 1 | + 0.894i·5-s − 0.755·7-s + 1.20i·11-s − 0.554i·13-s + 0.970·17-s + 0.917i·19-s + 1.66·23-s + 0.200·25-s − 1.11i·29-s − 1.07·31-s − 0.676i·35-s − 0.328i·37-s − 1.87·41-s + 1.82i·43-s − 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.128267099\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.128267099\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 2iT - 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 - 12iT - 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 8iT - 59T^{2} \) |
| 61 | \( 1 - 10iT - 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 14T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 8T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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
−9.542589884284167678380497047633, −8.492439858865446723316560468248, −7.50878479422072028238821596199, −7.09322719693834748955918835130, −6.25496000786573372800596195797, −5.45701462269163133544264688803, −4.45085927168261632939298814040, −3.32731163603403322155229686646, −2.83041310730183062747333376547, −1.46846804517886797743116627437,
0.39876046015446341899127553863, 1.53562119823161585881403749417, 3.11604440093850998158291949664, 3.57356561707363237351944766725, 5.09522196778896633967211852956, 5.21667449640218402278133474906, 6.54042905847488682980164784222, 6.98078916625179322064920091345, 8.152036909016030893130358617456, 8.876823183834166789100568286911