L(s) = 1 | + (1.20 + 2.09i)2-s + (−0.5 + 0.866i)3-s + (−1.91 + 3.31i)4-s + (−0.292 − 0.507i)5-s − 2.41·6-s − 4.41·8-s + (−0.499 − 0.866i)9-s + (0.707 − 1.22i)10-s + (1 − 1.73i)11-s + (−1.91 − 3.31i)12-s + 5.41·13-s + 0.585·15-s + (−1.49 − 2.59i)16-s + (−3.12 + 5.40i)17-s + (1.20 − 2.09i)18-s + (−1.41 − 2.44i)19-s + ⋯ |
L(s) = 1 | + (0.853 + 1.47i)2-s + (−0.288 + 0.499i)3-s + (−0.957 + 1.65i)4-s + (−0.130 − 0.226i)5-s − 0.985·6-s − 1.56·8-s + (−0.166 − 0.288i)9-s + (0.223 − 0.387i)10-s + (0.301 − 0.522i)11-s + (−0.552 − 0.957i)12-s + 1.50·13-s + 0.151·15-s + (−0.374 − 0.649i)16-s + (−0.757 + 1.31i)17-s + (0.284 − 0.492i)18-s + (−0.324 − 0.561i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.518856 + 1.37226i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.518856 + 1.37226i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-1.20 - 2.09i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (0.292 + 0.507i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 5.41T + 13T^{2} \) |
| 17 | \( 1 + (3.12 - 5.40i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.41 + 2.44i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.82 + 3.16i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 1.17T + 29T^{2} \) |
| 31 | \( 1 + (-3.41 + 5.91i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 2.24T + 41T^{2} \) |
| 43 | \( 1 + 5.65T + 43T^{2} \) |
| 47 | \( 1 + (1.41 + 2.44i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1 + 1.73i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.41 + 5.91i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.87 + 3.25i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.82 - 4.89i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 + (-2.94 + 5.10i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.17 + 2.02i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 15.3T + 83T^{2} \) |
| 89 | \( 1 + (-2.87 - 4.98i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 5.41T + 97T^{2} \) |
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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
−13.53251751886143408672240391193, −12.93009963273447018517360187956, −11.58404576005601933645857906170, −10.49193099477794836745679679778, −8.734890741513374116950316770026, −8.248126502192753607369006305068, −6.51096084429568302594207450597, −6.04602224659697085485975387550, −4.61891468562083640729430301363, −3.75065186391036688153790662991,
1.55290516222439578912540029753, 3.16040508286271385642267829764, 4.45176021106498281617327033521, 5.77188774024050862575162382151, 7.10883485010791613238355851647, 8.804703681731128566029989309013, 10.04373210195677567601663104424, 11.11452889683022765748305111668, 11.61506682813933755733797129145, 12.59816341036391666968585424405