L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (−0.606 − 0.350i)5-s − 0.999·6-s + (−1.82 − 1.91i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (0.350 + 0.606i)10-s + (2.78 − 1.80i)11-s + (0.866 + 0.499i)12-s − 7.03·13-s + (0.629 + 2.56i)14-s − 0.700·15-s + (−0.5 + 0.866i)16-s + (−0.308 − 0.535i)17-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.271 − 0.156i)5-s − 0.408·6-s + (−0.691 − 0.722i)7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (0.110 + 0.191i)10-s + (0.838 − 0.545i)11-s + (0.249 + 0.144i)12-s − 1.95·13-s + (0.168 + 0.686i)14-s − 0.180·15-s + (−0.125 + 0.216i)16-s + (−0.0749 − 0.129i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.771 + 0.636i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.771 + 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.266912 - 0.742483i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.266912 - 0.742483i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 7 | \( 1 + (1.82 + 1.91i)T \) |
| 11 | \( 1 + (-2.78 + 1.80i)T \) |
good | 5 | \( 1 + (0.606 + 0.350i)T + (2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 + 7.03T + 13T^{2} \) |
| 17 | \( 1 + (0.308 + 0.535i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.391 + 0.678i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.63 + 4.56i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 2.09iT - 29T^{2} \) |
| 31 | \( 1 + (5.35 - 3.08i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.99 + 3.45i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 5.85T + 41T^{2} \) |
| 43 | \( 1 - 3.62iT - 43T^{2} \) |
| 47 | \( 1 + (2.03 + 1.17i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.95 + 6.84i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.25 - 1.88i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.18 + 7.24i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.07 + 1.85i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 4.48T + 71T^{2} \) |
| 73 | \( 1 + (-2.64 - 4.58i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.75 - 2.16i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 4.03T + 83T^{2} \) |
| 89 | \( 1 + (-14.0 - 8.10i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 10.4iT - 97T^{2} \) |
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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
−10.55188072407130131086823935485, −9.655602941084177371475444964326, −9.096133775729282180724661866380, −7.960744810613905135009168448093, −7.21088955969433828218434041917, −6.40570721020424639595934531414, −4.63523000163807857464402266694, −3.51094534076612241702652944301, −2.36549013963713380528654773343, −0.53711208981894313225437785757,
2.08059847909967549534933193485, 3.32958550794924235812631679927, 4.74843609083530323717746496681, 5.88034575608749338715307804707, 7.15134177329453624137160624626, 7.59870882766104034941505086270, 9.004107937629513127929797883370, 9.446073922089749470237561003012, 10.09526657527100050990229122222, 11.36754562393672871103239686568