| L(s) = 1 | + (1.32 + 1.11i)3-s − 2.23i·5-s + 2.64·7-s + (0.5 + 2.95i)9-s + 5.91i·11-s + 2.64·13-s + (2.50 − 2.95i)15-s + 2.23i·17-s + (3.50 + 2.95i)21-s − 5.00·25-s + (−2.64 + 4.47i)27-s − 5.91i·29-s + (−6.61 + 7.82i)33-s − 5.91i·35-s + (3.50 + 2.95i)39-s + ⋯ |
| L(s) = 1 | + (0.763 + 0.645i)3-s − 0.999i·5-s + 0.999·7-s + (0.166 + 0.986i)9-s + 1.78i·11-s + 0.733·13-s + (0.645 − 0.763i)15-s + 0.542i·17-s + (0.763 + 0.645i)21-s − 1.00·25-s + (−0.509 + 0.860i)27-s − 1.09i·29-s + (−1.15 + 1.36i)33-s − 0.999i·35-s + (0.560 + 0.473i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.645 - 0.763i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.645 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.575867497\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.575867497\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.32 - 1.11i)T \) |
| 5 | \( 1 + 2.23iT \) |
| 7 | \( 1 - 2.64T \) |
| good | 11 | \( 1 - 5.91iT - 11T^{2} \) |
| 13 | \( 1 - 2.64T + 13T^{2} \) |
| 17 | \( 1 - 2.23iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 5.91iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 11.1iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 11.8iT - 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 - T + 79T^{2} \) |
| 83 | \( 1 + 8.94iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 18.5T + 97T^{2} \) |
| 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
−9.379848192735294361021552518228, −8.720386269714447252084914173897, −7.939351269325077570501980897889, −7.51081141673473194106519147718, −6.09812943164351163838147747091, −4.97478162678790030623157479396, −4.52897198952366180722656194167, −3.81058416471280730985409857525, −2.25620305537984466720236272901, −1.50504930325271593128499519220,
0.987091973113866678190838040365, 2.22092022367522043124687045394, 3.21284286801102683484996634165, 3.81300495629830335409044942467, 5.31712709551524249238507716730, 6.17222088708926379418825205322, 6.93633576982369294721554630260, 7.71744351816188103250288745195, 8.478320022035703595871627067991, 8.875575741355872677074686417395
