| L(s) = 1 | + (−2.96 − 2.36i)2-s + (−2.18 + 1.73i)3-s + (2.30 + 10.0i)4-s + (0.910 − 1.89i)5-s + 10.5·6-s + 8.82i·7-s + (10.4 − 21.7i)8-s + (−0.270 + 1.18i)9-s + (−7.16 + 3.44i)10-s + (−3.64 + 15.9i)11-s + (−22.5 − 18.0i)12-s + (−12.2 − 5.91i)13-s + (20.8 − 26.1i)14-s + (1.30 + 5.70i)15-s + (−44.9 + 21.6i)16-s + (−9.25 + 4.45i)17-s + ⋯ |
| L(s) = 1 | + (−1.48 − 1.18i)2-s + (−0.727 + 0.579i)3-s + (0.576 + 2.52i)4-s + (0.182 − 0.378i)5-s + 1.76·6-s + 1.26i·7-s + (1.30 − 2.71i)8-s + (−0.0300 + 0.131i)9-s + (−0.716 + 0.344i)10-s + (−0.331 + 1.45i)11-s + (−1.88 − 1.50i)12-s + (−0.945 − 0.455i)13-s + (1.48 − 1.86i)14-s + (0.0868 + 0.380i)15-s + (−2.80 + 1.35i)16-s + (−0.544 + 0.262i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 - 0.855i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.518 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.298533 + 0.168158i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.298533 + 0.168158i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-42.1 - 8.63i)T \) |
| good | 2 | \( 1 + (2.96 + 2.36i)T + (0.890 + 3.89i)T^{2} \) |
| 3 | \( 1 + (2.18 - 1.73i)T + (2.00 - 8.77i)T^{2} \) |
| 5 | \( 1 + (-0.910 + 1.89i)T + (-15.5 - 19.5i)T^{2} \) |
| 7 | \( 1 - 8.82iT - 49T^{2} \) |
| 11 | \( 1 + (3.64 - 15.9i)T + (-109. - 52.4i)T^{2} \) |
| 13 | \( 1 + (12.2 + 5.91i)T + (105. + 132. i)T^{2} \) |
| 17 | \( 1 + (9.25 - 4.45i)T + (180. - 225. i)T^{2} \) |
| 19 | \( 1 + (-22.6 + 5.16i)T + (325. - 156. i)T^{2} \) |
| 23 | \( 1 + (-2.32 + 10.1i)T + (-476. - 229. i)T^{2} \) |
| 29 | \( 1 + (6.41 + 5.11i)T + (187. + 819. i)T^{2} \) |
| 31 | \( 1 + (17.8 - 22.3i)T + (-213. - 936. i)T^{2} \) |
| 37 | \( 1 - 11.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-17.4 + 21.8i)T + (-374. - 1.63e3i)T^{2} \) |
| 47 | \( 1 + (-8.30 - 36.3i)T + (-1.99e3 + 958. i)T^{2} \) |
| 53 | \( 1 + (-8.13 + 3.91i)T + (1.75e3 - 2.19e3i)T^{2} \) |
| 59 | \( 1 + (4.20 - 2.02i)T + (2.17e3 - 2.72e3i)T^{2} \) |
| 61 | \( 1 + (-44.0 + 35.1i)T + (828. - 3.62e3i)T^{2} \) |
| 67 | \( 1 + (4.85 + 21.2i)T + (-4.04e3 + 1.94e3i)T^{2} \) |
| 71 | \( 1 + (88.9 - 20.2i)T + (4.54e3 - 2.18e3i)T^{2} \) |
| 73 | \( 1 + (-14.3 + 29.7i)T + (-3.32e3 - 4.16e3i)T^{2} \) |
| 79 | \( 1 - 113.T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-62.6 - 78.5i)T + (-1.53e3 + 6.71e3i)T^{2} \) |
| 89 | \( 1 + (-24.0 + 19.1i)T + (1.76e3 - 7.72e3i)T^{2} \) |
| 97 | \( 1 + (4.29 - 18.7i)T + (-8.47e3 - 4.08e3i)T^{2} \) |
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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
−16.33580836306471899756022905209, −15.33046720089714510930153708320, −12.72490316806998428255822019306, −12.08607673656673672416104884072, −10.93534514033745673772199302465, −9.859602695522800987280165092499, −9.056943877149334241017183891888, −7.54121879208800799499424645563, −5.01839806970440115128118263531, −2.37604543734515143268444538724,
0.64297277833505420136453903397, 5.66773376406517206217395354671, 6.84550474771072393518449096627, 7.62338667952256929968136942729, 9.246639987000108179109340214058, 10.50478300462224766714034141541, 11.43291725478410591994487990193, 13.65438849278896570624866480379, 14.60935553283386585933782682240, 16.17998796166904016683963914303
