L(s) = 1 | + (−0.866 + 1.5i)2-s + (−0.5 − 0.866i)4-s + (−1.73 + 3i)5-s − 1.73·8-s + (−3 − 5.19i)10-s + (−1.73 − 3i)11-s − 2·13-s + (2.49 − 4.33i)16-s + (1.73 + 3i)17-s + (−2 + 3.46i)19-s + 3.46·20-s + 6·22-s + (1.73 − 3i)23-s + (−3.5 − 6.06i)25-s + (1.73 − 3i)26-s + ⋯ |
L(s) = 1 | + (−0.612 + 1.06i)2-s + (−0.250 − 0.433i)4-s + (−0.774 + 1.34i)5-s − 0.612·8-s + (−0.948 − 1.64i)10-s + (−0.522 − 0.904i)11-s − 0.554·13-s + (0.624 − 1.08i)16-s + (0.420 + 0.727i)17-s + (−0.458 + 0.794i)19-s + 0.774·20-s + 1.27·22-s + (0.361 − 0.625i)23-s + (−0.700 − 1.21i)25-s + (0.339 − 0.588i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.173956 - 0.228662i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.173956 - 0.228662i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.866 - 1.5i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (1.73 - 3i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.73 + 3i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + (-1.73 - 3i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.73 + 3i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (-3.46 + 6i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.46 - 6i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.46 + 6i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + (-7 - 12.1i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (1.73 - 3i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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
−11.63145953140864884712330875141, −10.67481477603085107661468994950, −10.00940385348994075452426782494, −8.601052944166627185289607522039, −8.026184959205837956307271114123, −7.20646216331169981442250147870, −6.46813622227801216237902914166, −5.53650651591020558543008435591, −3.76222455969274628938816775019, −2.78397193683996277240462990076,
0.21389328570640586547052829572, 1.70010740464909388783429780781, 3.13026816316691568427336711101, 4.55036729400383037111854860421, 5.32031628841393652010864535379, 7.04237826403988946377928014974, 8.032540759862243909812779302945, 8.954840316107301909266339496171, 9.556841868007172822792519837087, 10.47264876708145142538978607657