L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + 4.27·5-s + (−0.499 − 0.866i)6-s + (−0.5 − 0.866i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (−2.13 + 3.70i)10-s + (2 − 3.46i)11-s + 0.999·12-s + (3.5 + 0.866i)13-s + 0.999·14-s + (−2.13 + 3.70i)15-s + (−0.5 + 0.866i)16-s + (−1.5 − 2.59i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + 1.91·5-s + (−0.204 − 0.353i)6-s + (−0.188 − 0.327i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.675 + 1.17i)10-s + (0.603 − 1.04i)11-s + 0.288·12-s + (0.970 + 0.240i)13-s + 0.267·14-s + (−0.551 + 0.955i)15-s + (−0.125 + 0.216i)16-s + (−0.363 − 0.630i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.45575 + 0.400089i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.45575 + 0.400089i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 + (-3.5 - 0.866i)T \) |
good | 5 | \( 1 - 4.27T + 5T^{2} \) |
| 11 | \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.27 + 5.67i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.63 + 4.56i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.13 - 7.16i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 1.27T + 31T^{2} \) |
| 37 | \( 1 + (0.137 - 0.238i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.13 - 7.16i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.91 - 10.2i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 6.54T + 47T^{2} \) |
| 53 | \( 1 + 3.54T + 53T^{2} \) |
| 59 | \( 1 + (-5.91 - 10.2i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.5 - 7.79i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.36 + 2.35i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2.63 + 4.56i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 5.72T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 + (0.362 - 0.627i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.274 - 0.476i)T + (-48.5 + 84.0i)T^{2} \) |
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
−10.81612686651108537584171496689, −9.829748268604782101754423572577, −9.023375100545003505405037207302, −8.712081039041671103209102551350, −6.78993651485837534191489007827, −6.39246210159131191168886133129, −5.53192785511085364996752855248, −4.55626558573150551997766725923, −2.90541959188312581433680185504, −1.20874050595958855215314506543,
1.62184992632624417131358898284, 2.12285541535609941235298984318, 3.80605076785381613302840167360, 5.39608951024021244020050534097, 6.09921726721841033672921502160, 6.93684760129807240353044800478, 8.321354610595611986598243092135, 9.196799336838843808314271882911, 9.890639418194349868791814596539, 10.56035633902108720902277889998