L(s) = 1 | + (−0.380 + 0.658i)2-s + (0.710 + 1.23i)4-s + (−1.59 + 2.75i)5-s + (−2.56 − 0.658i)7-s − 2.60·8-s + (−1.21 − 2.09i)10-s + (−1.11 − 1.93i)11-s + 3.70·13-s + (1.40 − 1.43i)14-s + (−0.430 + 0.746i)16-s + (2.80 + 4.85i)17-s + (−2.21 + 3.82i)19-s − 4.52·20-s + 1.70·22-s + (0.471 − 0.816i)23-s + ⋯ |
L(s) = 1 | + (−0.269 + 0.465i)2-s + (0.355 + 0.615i)4-s + (−0.711 + 1.23i)5-s + (−0.968 − 0.249i)7-s − 0.920·8-s + (−0.382 − 0.663i)10-s + (−0.337 − 0.584i)11-s + 1.02·13-s + (0.376 − 0.384i)14-s + (−0.107 + 0.186i)16-s + (0.679 + 1.17i)17-s + (−0.507 + 0.878i)19-s − 1.01·20-s + 0.363·22-s + (0.0982 − 0.170i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.757 - 0.653i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.757 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.278983 + 0.750354i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.278983 + 0.750354i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (2.56 + 0.658i)T \) |
good | 2 | \( 1 + (0.380 - 0.658i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (1.59 - 2.75i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.11 + 1.93i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 3.70T + 13T^{2} \) |
| 17 | \( 1 + (-2.80 - 4.85i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.21 - 3.82i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.471 + 0.816i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 10.1T + 29T^{2} \) |
| 31 | \( 1 + (-2.85 - 4.93i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.56 - 2.70i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 3.98T + 41T^{2} \) |
| 43 | \( 1 + 3.28T + 43T^{2} \) |
| 47 | \( 1 + (-0.112 + 0.195i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (5.33 + 9.23i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.02 - 1.78i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.92 + 5.05i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.71 + 6.42i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 7.26T + 71T^{2} \) |
| 73 | \( 1 + (3.77 + 6.54i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.41 + 5.91i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 8.11T + 83T^{2} \) |
| 89 | \( 1 + (4.86 - 8.42i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 0.842T + 97T^{2} \) |
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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
−12.81420294707857093075471484186, −11.98032509411326382501995732627, −10.85572345543298044306077048162, −10.22029121603948213788998953151, −8.526093124279887150220772190738, −7.903453258495766537455818247838, −6.62750669789568535006674279058, −6.22894900122790426762571758586, −3.69153561356004538361060102442, −3.08503916168646420411761682599,
0.789988307387670108050688681938, 2.83725192711501270840058828502, 4.52901464755819055874357286326, 5.75937009812810085312094954301, 6.97789496896642725653303542541, 8.436262809477058063217872831626, 9.286841450195114863319041773209, 10.11461738277046303281159931177, 11.32255209022962493210968773268, 12.13788996373488374467018924002