| L(s) = 1 | + i·2-s + i·3-s − 4-s + 1.58i·5-s − 6-s − i·8-s − 9-s − 1.58·10-s + (3.29 + 0.383i)11-s − i·12-s − 2.26·13-s − 1.58·15-s + 16-s + 2.99·17-s − i·18-s − 1.32·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.707i·5-s − 0.408·6-s − 0.353i·8-s − 0.333·9-s − 0.500·10-s + (0.993 + 0.115i)11-s − 0.288i·12-s − 0.629·13-s − 0.408·15-s + 0.250·16-s + 0.726·17-s − 0.235i·18-s − 0.304·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 + 0.300i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.953 + 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.330406482\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.330406482\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (-3.29 - 0.383i)T \) |
| good | 5 | \( 1 - 1.58iT - 5T^{2} \) |
| 13 | \( 1 + 2.26T + 13T^{2} \) |
| 17 | \( 1 - 2.99T + 17T^{2} \) |
| 19 | \( 1 + 1.32T + 19T^{2} \) |
| 23 | \( 1 - 2.32T + 23T^{2} \) |
| 29 | \( 1 - 2.90iT - 29T^{2} \) |
| 31 | \( 1 - 10.1iT - 31T^{2} \) |
| 37 | \( 1 + 2.83T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 - 12.0iT - 43T^{2} \) |
| 47 | \( 1 + 0.996iT - 47T^{2} \) |
| 53 | \( 1 + 0.531T + 53T^{2} \) |
| 59 | \( 1 - 0.121iT - 59T^{2} \) |
| 61 | \( 1 - 7.54T + 61T^{2} \) |
| 67 | \( 1 - 0.628T + 67T^{2} \) |
| 71 | \( 1 + 7.26T + 71T^{2} \) |
| 73 | \( 1 + 9.21T + 73T^{2} \) |
| 79 | \( 1 + 11.7iT - 79T^{2} \) |
| 83 | \( 1 + 7.48T + 83T^{2} \) |
| 89 | \( 1 - 11.3iT - 89T^{2} \) |
| 97 | \( 1 + 1.49iT - 97T^{2} \) |
| 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
−8.907915260488161986683168020543, −8.453244938036611605574191813130, −7.34897797901325510095827560205, −6.85436842868666691787304209479, −6.20133236540772014647966797104, −5.17218644058019197982191047312, −4.64451533032806826528712845291, −3.54366399871584015955331911134, −2.97267611781449520706920253530, −1.41493233202141523213499191248,
0.43496866770860771112998030201, 1.42844972210293591503448218722, 2.33361599531074096039265353780, 3.42542180113681459733075082935, 4.25350293143876121250809461120, 5.12753191196119873116999252803, 5.86224200473558896579048358548, 6.83081392874856072297378399301, 7.54658656067435348920284234629, 8.504539886080677465611206226269
