| L(s) = 1 | + (−0.290 − 0.231i)2-s + (−4.43 + 3.54i)3-s + (−0.859 − 3.76i)4-s + (−3.26 + 6.78i)5-s + 2.10·6-s + 1.66i·7-s + (−1.26 + 2.63i)8-s + (5.17 − 22.6i)9-s + (2.52 − 1.21i)10-s + (−2.00 + 8.79i)11-s + (17.1 + 13.6i)12-s + (9.47 + 4.56i)13-s + (0.386 − 0.484i)14-s + (−9.51 − 41.6i)15-s + (−12.9 + 6.23i)16-s + (2.62 − 1.26i)17-s + ⋯ |
| L(s) = 1 | + (−0.145 − 0.115i)2-s + (−1.47 + 1.18i)3-s + (−0.214 − 0.941i)4-s + (−0.653 + 1.35i)5-s + 0.351·6-s + 0.238i·7-s + (−0.158 + 0.328i)8-s + (0.574 − 2.51i)9-s + (0.252 − 0.121i)10-s + (−0.182 + 0.799i)11-s + (1.42 + 1.13i)12-s + (0.728 + 0.351i)13-s + (0.0275 − 0.0345i)14-s + (−0.634 − 2.77i)15-s + (−0.808 + 0.389i)16-s + (0.154 − 0.0743i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.758 - 0.651i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.758 - 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.140007 + 0.377855i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.140007 + 0.377855i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-15.4 + 40.1i)T \) |
| good | 2 | \( 1 + (0.290 + 0.231i)T + (0.890 + 3.89i)T^{2} \) |
| 3 | \( 1 + (4.43 - 3.54i)T + (2.00 - 8.77i)T^{2} \) |
| 5 | \( 1 + (3.26 - 6.78i)T + (-15.5 - 19.5i)T^{2} \) |
| 7 | \( 1 - 1.66iT - 49T^{2} \) |
| 11 | \( 1 + (2.00 - 8.79i)T + (-109. - 52.4i)T^{2} \) |
| 13 | \( 1 + (-9.47 - 4.56i)T + (105. + 132. i)T^{2} \) |
| 17 | \( 1 + (-2.62 + 1.26i)T + (180. - 225. i)T^{2} \) |
| 19 | \( 1 + (14.6 - 3.33i)T + (325. - 156. i)T^{2} \) |
| 23 | \( 1 + (4.92 - 21.5i)T + (-476. - 229. i)T^{2} \) |
| 29 | \( 1 + (-9.07 - 7.24i)T + (187. + 819. i)T^{2} \) |
| 31 | \( 1 + (10.0 - 12.5i)T + (-213. - 936. i)T^{2} \) |
| 37 | \( 1 - 2.89iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-36.6 + 46.0i)T + (-374. - 1.63e3i)T^{2} \) |
| 47 | \( 1 + (-6.80 - 29.7i)T + (-1.99e3 + 958. i)T^{2} \) |
| 53 | \( 1 + (53.8 - 25.9i)T + (1.75e3 - 2.19e3i)T^{2} \) |
| 59 | \( 1 + (30.4 - 14.6i)T + (2.17e3 - 2.72e3i)T^{2} \) |
| 61 | \( 1 + (-40.1 + 32.0i)T + (828. - 3.62e3i)T^{2} \) |
| 67 | \( 1 + (4.65 + 20.4i)T + (-4.04e3 + 1.94e3i)T^{2} \) |
| 71 | \( 1 + (18.2 - 4.17i)T + (4.54e3 - 2.18e3i)T^{2} \) |
| 73 | \( 1 + (60.1 - 124. i)T + (-3.32e3 - 4.16e3i)T^{2} \) |
| 79 | \( 1 - 62.6T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-77.8 - 97.5i)T + (-1.53e3 + 6.71e3i)T^{2} \) |
| 89 | \( 1 + (29.2 - 23.3i)T + (1.76e3 - 7.72e3i)T^{2} \) |
| 97 | \( 1 + (18.5 - 81.2i)T + (-8.47e3 - 4.08e3i)T^{2} \) |
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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
−15.81707419440554255830621864612, −15.34987232558981976237774430279, −14.37568673953102327868880033994, −12.08898457980393271236960694233, −10.96896349708549139356982569337, −10.56430807964693805782891561330, −9.401320822343714888718251047084, −6.78391088760592452957097962352, −5.63351693832223669752250502790, −4.09302866586312131142199969255,
0.55689676403349338025771368850, 4.54963452442674093553676893833, 6.15162841135985318790742029954, 7.72250648520364778860143101453, 8.522058854429507219398394285761, 10.98172511338942601991081979172, 11.98419217774595578069029405741, 12.80627992267915486157581099991, 13.36506372693225332154275254575, 16.15533068904315698013663948813
