L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.500 − 0.866i)4-s + (−1 − 1.73i)5-s − 3·8-s + (−0.999 + 1.73i)10-s + (2 − 3.46i)11-s − 2·13-s + (0.500 + 0.866i)16-s + (−3 + 5.19i)17-s + (−2 − 3.46i)19-s − 2·20-s − 3.99·22-s + (0.500 − 0.866i)25-s + (1 + 1.73i)26-s + 2·29-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.250 − 0.433i)4-s + (−0.447 − 0.774i)5-s − 1.06·8-s + (−0.316 + 0.547i)10-s + (0.603 − 1.04i)11-s − 0.554·13-s + (0.125 + 0.216i)16-s + (−0.727 + 1.26i)17-s + (−0.458 − 0.794i)19-s − 0.447·20-s − 0.852·22-s + (0.100 − 0.173i)25-s + (0.196 + 0.339i)26-s + 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0528210 - 0.832349i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0528210 - 0.832349i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.5 + 0.866i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3 + 5.19i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-3 + 5.19i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-8 - 13.8i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + (7 + 12.1i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 18T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.81816057609690301350209065938, −9.863927074629101212979322956917, −8.785828459576528146697069600297, −8.448891794333314692240576878529, −6.85542351953130888969222340940, −5.98548577948485881999583093687, −4.79208794793551421877151840854, −3.55528996331510143687911431437, −2.06372222323479133124052619410, −0.56574615748523627800600037904,
2.38234167632467782041601104852, 3.56093595562485365691255699573, 4.82861947041117472738471645570, 6.38833289892065249201816713922, 7.05056013115321086583392089416, 7.63060349210109494651241503445, 8.736676061247533233536899379302, 9.613997248842167876723362570465, 10.61409736355128903825277258672, 11.80402897285165985849555844608