| L(s) = 1 | − 2i·5-s − 7-s − 4i·11-s + 2i·13-s − 2·17-s − 6·23-s + 25-s − 2i·29-s − 4·31-s + 2i·35-s + 4i·37-s − 10·41-s − 2i·43-s + 8·47-s + 49-s + ⋯ |
| L(s) = 1 | − 0.894i·5-s − 0.377·7-s − 1.20i·11-s + 0.554i·13-s − 0.485·17-s − 1.25·23-s + 0.200·25-s − 0.371i·29-s − 0.718·31-s + 0.338i·35-s + 0.657i·37-s − 1.56·41-s − 0.304i·43-s + 1.16·47-s + 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + 2iT - 5T^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 - 10iT - 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 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
−8.153735978261578223385960291711, −7.28293790534255113927349713495, −6.37482047582959026842693795905, −5.81537083539061278952887681628, −4.97912670621564574849739580713, −4.15140229843289014990721759156, −3.40931834227336352461453488970, −2.30236995665996389646233400656, −1.18453570702750091851084579796, 0,
1.76699977821076130376209170245, 2.58817032182108882884166022684, 3.48566695571550830501192859846, 4.27221871920317547065621442920, 5.22255497649759042214106801196, 6.03672883668108990015910097482, 6.88261246340586586090784656541, 7.22566376029179505182743829402, 8.094695036593806376470951558547
