| L(s) = 1 | + (−0.311 + 0.723i)2-s + (0.946 + 1.00i)4-s + (−0.161 − 2.78i)5-s + (4.84 + 1.14i)7-s + (−2.50 + 0.910i)8-s + (2.06 + 0.750i)10-s + (1.45 + 0.954i)11-s + (−2.15 − 0.252i)13-s + (−2.34 + 3.14i)14-s + (−0.0385 + 0.662i)16-s + (3.39 − 2.84i)17-s + (−1.63 − 1.37i)19-s + (2.63 − 2.79i)20-s + (−1.14 + 0.751i)22-s + (0.659 − 0.156i)23-s + ⋯ |
| L(s) = 1 | + (−0.220 + 0.511i)2-s + (0.473 + 0.501i)4-s + (−0.0724 − 1.24i)5-s + (1.83 + 0.434i)7-s + (−0.884 + 0.321i)8-s + (0.651 + 0.237i)10-s + (0.437 + 0.287i)11-s + (−0.598 − 0.0699i)13-s + (−0.626 + 0.841i)14-s + (−0.00964 + 0.165i)16-s + (0.823 − 0.690i)17-s + (−0.374 − 0.314i)19-s + (0.589 − 0.625i)20-s + (−0.243 + 0.160i)22-s + (0.137 − 0.0326i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.576i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 - 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.73396 + 0.550540i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.73396 + 0.550540i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (0.311 - 0.723i)T + (-1.37 - 1.45i)T^{2} \) |
| 5 | \( 1 + (0.161 + 2.78i)T + (-4.96 + 0.580i)T^{2} \) |
| 7 | \( 1 + (-4.84 - 1.14i)T + (6.25 + 3.14i)T^{2} \) |
| 11 | \( 1 + (-1.45 - 0.954i)T + (4.35 + 10.1i)T^{2} \) |
| 13 | \( 1 + (2.15 + 0.252i)T + (12.6 + 2.99i)T^{2} \) |
| 17 | \( 1 + (-3.39 + 2.84i)T + (2.95 - 16.7i)T^{2} \) |
| 19 | \( 1 + (1.63 + 1.37i)T + (3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (-0.659 + 0.156i)T + (20.5 - 10.3i)T^{2} \) |
| 29 | \( 1 + (-3.43 - 4.61i)T + (-8.31 + 27.7i)T^{2} \) |
| 31 | \( 1 + (-1.68 + 5.63i)T + (-25.9 - 17.0i)T^{2} \) |
| 37 | \( 1 + (-0.131 - 0.747i)T + (-34.7 + 12.6i)T^{2} \) |
| 41 | \( 1 + (0.0489 + 0.113i)T + (-28.1 + 29.8i)T^{2} \) |
| 43 | \( 1 + (-2.27 + 1.14i)T + (25.6 - 34.4i)T^{2} \) |
| 47 | \( 1 + (-0.487 - 1.62i)T + (-39.2 + 25.8i)T^{2} \) |
| 53 | \( 1 + (-5.02 - 8.69i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (9.71 - 6.39i)T + (23.3 - 54.1i)T^{2} \) |
| 61 | \( 1 + (5.43 - 5.76i)T + (-3.54 - 60.8i)T^{2} \) |
| 67 | \( 1 + (0.277 - 0.373i)T + (-19.2 - 64.1i)T^{2} \) |
| 71 | \( 1 + (11.4 + 4.17i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (2.01 - 0.732i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-2.77 + 6.43i)T + (-54.2 - 57.4i)T^{2} \) |
| 83 | \( 1 + (-1.31 + 3.03i)T + (-56.9 - 60.3i)T^{2} \) |
| 89 | \( 1 + (-4.72 + 1.71i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-0.436 + 7.49i)T + (-96.3 - 11.2i)T^{2} \) |
| 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
−10.54338009968055484658084145499, −9.087577568259890177558967597489, −8.769913147225690789080328245485, −7.80021353086749705331753108112, −7.39546801670945308877338747240, −5.95607170604455607481620992163, −5.02531176782677750305130825678, −4.38990677185950547343406878116, −2.62764476389374598748075867448, −1.34984444609586681311894545501,
1.33158102871440752767180549102, 2.36345724251961030076923973096, 3.55963777811334141435665735538, 4.82612562424642618293196761236, 5.95745629781951439756894486555, 6.85873283202547520631927207797, 7.68952922400875068986823863106, 8.532952894251124688627573259644, 9.842644972684114658839108437295, 10.54990973732059735501191635826
