| L(s) = 1 | + (−0.983 − 1.01i)2-s + (−1.72 + 0.160i)3-s + (−0.0664 + 1.99i)4-s + (−0.805 + 2.21i)5-s + (1.85 + 1.59i)6-s + (0.984 + 0.173i)7-s + (2.09 − 1.89i)8-s + (2.94 − 0.554i)9-s + (3.04 − 1.35i)10-s + (1.94 − 0.709i)11-s + (−0.206 − 3.45i)12-s + (−3.19 + 2.67i)13-s + (−0.791 − 1.17i)14-s + (1.03 − 3.94i)15-s + (−3.99 − 0.265i)16-s + (4.68 − 2.70i)17-s + ⋯ |
| L(s) = 1 | + (−0.695 − 0.718i)2-s + (−0.995 + 0.0928i)3-s + (−0.0332 + 0.999i)4-s + (−0.360 + 0.989i)5-s + (0.758 + 0.651i)6-s + (0.372 + 0.0656i)7-s + (0.741 − 0.671i)8-s + (0.982 − 0.184i)9-s + (0.961 − 0.429i)10-s + (0.587 − 0.213i)11-s + (−0.0597 − 0.998i)12-s + (−0.885 + 0.743i)13-s + (−0.211 − 0.313i)14-s + (0.266 − 1.01i)15-s + (−0.997 − 0.0664i)16-s + (1.13 − 0.655i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.160 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.160 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.435825 + 0.370553i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.435825 + 0.370553i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.983 + 1.01i)T \) |
| 3 | \( 1 + (1.72 - 0.160i)T \) |
| 7 | \( 1 + (-0.984 - 0.173i)T \) |
| good | 5 | \( 1 + (0.805 - 2.21i)T + (-3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (-1.94 + 0.709i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (3.19 - 2.67i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-4.68 + 2.70i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.343 - 0.198i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.240 - 1.36i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-2.74 + 3.26i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (9.24 - 1.62i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-5.36 - 9.28i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (6.40 + 7.63i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.80 - 7.69i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (1.63 - 9.26i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 5.29iT - 53T^{2} \) |
| 59 | \( 1 + (-1.36 - 0.496i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (1.74 - 9.87i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (1.72 + 2.05i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (5.21 + 9.02i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.42 + 5.93i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.98 - 7.12i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-7.37 - 6.18i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-6.47 - 3.73i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.37 - 0.499i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.67022269583391995684046577204, −9.820967974661620339851803543387, −9.197659451668965517838051239541, −7.75492662060748525679445998838, −7.23291755514447419694493087179, −6.36568312372928602638591507905, −5.00372418656151742118218060788, −3.96888708505857568425134188926, −2.86866762953614883149582144757, −1.34437180467724322264487529799,
0.47689114271664008583664033358, 1.64702162528261673428576569103, 4.09614364102852790069679304139, 5.15109361980401370665611399409, 5.55518145273604552546906106056, 6.77587613870222494322453765979, 7.57531721506881133243879004903, 8.292180263636246521413142087722, 9.282607468456395730617848194442, 10.10213195621728113342099085688
