L(s) = 1 | + (0.247 − 0.429i)2-s + (0.877 + 1.51i)4-s + (1.84 − 3.19i)5-s + (1.68 + 2.04i)7-s + 1.86·8-s + (−0.915 − 1.58i)10-s + (0.446 + 0.772i)11-s − 1.19·13-s + (1.29 − 0.216i)14-s + (−1.29 + 2.23i)16-s + (−0.124 − 0.216i)17-s + (1.40 − 2.43i)19-s + 6.47·20-s + 0.442·22-s + (−1.23 + 2.14i)23-s + ⋯ |
L(s) = 1 | + (0.175 − 0.303i)2-s + (0.438 + 0.759i)4-s + (0.825 − 1.43i)5-s + (0.636 + 0.771i)7-s + 0.658·8-s + (−0.289 − 0.501i)10-s + (0.134 + 0.233i)11-s − 0.331·13-s + (0.345 − 0.0578i)14-s + (−0.323 + 0.559i)16-s + (−0.0303 − 0.0525i)17-s + (0.322 − 0.557i)19-s + 1.44·20-s + 0.0943·22-s + (−0.258 + 0.447i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.14590 - 0.318233i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.14590 - 0.318233i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-1.68 - 2.04i)T \) |
good | 2 | \( 1 + (-0.247 + 0.429i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (-1.84 + 3.19i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.446 - 0.772i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 1.19T + 13T^{2} \) |
| 17 | \( 1 + (0.124 + 0.216i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.40 + 2.43i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.23 - 2.14i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 4.14T + 29T^{2} \) |
| 31 | \( 1 + (1.79 + 3.10i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.36 - 4.09i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 4.78T + 41T^{2} \) |
| 43 | \( 1 - 9.97T + 43T^{2} \) |
| 47 | \( 1 + (-5.08 + 8.81i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.94 + 8.56i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.906 + 1.56i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.40 - 9.35i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.514 + 0.891i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 4.94T + 71T^{2} \) |
| 73 | \( 1 + (0.915 + 1.58i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.899 + 1.55i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 89 | \( 1 + (1.20 - 2.08i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
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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
−10.89547725973241553117055480119, −9.615656902855018672546922713636, −8.985314971701460495757305647820, −8.177984890292734661376935446608, −7.26721204101600807103288791006, −5.86797822489577958818047326051, −5.07505186410702096750933535820, −4.14484042475261234181761442525, −2.52363969580204646353139037546, −1.58706660194639455794579954313,
1.58134035456493381990220559849, 2.74246329292561161040652167153, 4.20802142504761672680008702949, 5.54844835501179322555718763912, 6.19898055204714774311799313277, 7.15368210802432678637470835064, 7.66250502125030668063762218026, 9.309256767419012659863718567117, 10.12764305275506122048412111670, 10.87810189219608019783881362757