| L(s) = 1 | + (−0.976 − 1.22i)2-s + (−3.38 + 4.24i)3-s + (1.23 − 5.40i)4-s + (−7.32 + 3.52i)5-s + 8.49·6-s − 30.7·7-s + (−19.1 + 9.20i)8-s + (−0.538 − 2.36i)9-s + (11.4 + 5.52i)10-s + (−2.01 − 8.83i)11-s + (18.7 + 23.5i)12-s + (14.5 − 6.98i)13-s + (30.0 + 37.6i)14-s + (9.80 − 42.9i)15-s + (−10.0 − 4.83i)16-s + (12.0 + 5.82i)17-s + ⋯ |
| L(s) = 1 | + (−0.345 − 0.432i)2-s + (−0.650 + 0.816i)3-s + (0.154 − 0.675i)4-s + (−0.654 + 0.315i)5-s + 0.578·6-s − 1.65·7-s + (−0.844 + 0.406i)8-s + (−0.0199 − 0.0874i)9-s + (0.362 + 0.174i)10-s + (−0.0552 − 0.242i)11-s + (0.451 + 0.565i)12-s + (0.309 − 0.149i)13-s + (0.572 + 0.718i)14-s + (0.168 − 0.739i)15-s + (−0.156 − 0.0755i)16-s + (0.172 + 0.0831i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.287i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.957 - 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.00724330 + 0.0493992i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00724330 + 0.0493992i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (279. + 39.2i)T \) |
| good | 2 | \( 1 + (0.976 + 1.22i)T + (-1.78 + 7.79i)T^{2} \) |
| 3 | \( 1 + (3.38 - 4.24i)T + (-6.00 - 26.3i)T^{2} \) |
| 5 | \( 1 + (7.32 - 3.52i)T + (77.9 - 97.7i)T^{2} \) |
| 7 | \( 1 + 30.7T + 343T^{2} \) |
| 11 | \( 1 + (2.01 + 8.83i)T + (-1.19e3 + 577. i)T^{2} \) |
| 13 | \( 1 + (-14.5 + 6.98i)T + (1.36e3 - 1.71e3i)T^{2} \) |
| 17 | \( 1 + (-12.0 - 5.82i)T + (3.06e3 + 3.84e3i)T^{2} \) |
| 19 | \( 1 + (-17.0 + 74.7i)T + (-6.17e3 - 2.97e3i)T^{2} \) |
| 23 | \( 1 + (-8.55 - 37.4i)T + (-1.09e4 + 5.27e3i)T^{2} \) |
| 29 | \( 1 + (-156. - 196. i)T + (-5.42e3 + 2.37e4i)T^{2} \) |
| 31 | \( 1 + (75.2 + 94.4i)T + (-6.62e3 + 2.90e4i)T^{2} \) |
| 37 | \( 1 + 391.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (211. + 264. i)T + (-1.53e4 + 6.71e4i)T^{2} \) |
| 47 | \( 1 + (26.7 - 117. i)T + (-9.35e4 - 4.50e4i)T^{2} \) |
| 53 | \( 1 + (-433. - 208. i)T + (9.28e4 + 1.16e5i)T^{2} \) |
| 59 | \( 1 + (322. + 155. i)T + (1.28e5 + 1.60e5i)T^{2} \) |
| 61 | \( 1 + (369. - 463. i)T + (-5.05e4 - 2.21e5i)T^{2} \) |
| 67 | \( 1 + (-149. + 655. i)T + (-2.70e5 - 1.30e5i)T^{2} \) |
| 71 | \( 1 + (93.0 - 407. i)T + (-3.22e5 - 1.55e5i)T^{2} \) |
| 73 | \( 1 + (-831. + 400. i)T + (2.42e5 - 3.04e5i)T^{2} \) |
| 79 | \( 1 + 107.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (625. - 784. i)T + (-1.27e5 - 5.57e5i)T^{2} \) |
| 89 | \( 1 + (117. - 147. i)T + (-1.56e5 - 6.87e5i)T^{2} \) |
| 97 | \( 1 + (107. + 469. i)T + (-8.22e5 + 3.95e5i)T^{2} \) |
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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
−15.75238657290183754272536722507, −15.43589855162024387296221239883, −13.63271291768999985096244088623, −12.07510081704742677799612626995, −10.91119316584577376481195812361, −10.19212931145296111728397356842, −9.096214332898880510816060958237, −6.81172816591266449063452318104, −5.42710626364945978412033643730, −3.36004993566397471027153350886,
0.04603071787579590910541255792, 3.54286736861492852134174783599, 6.23264215299408084551911064429, 7.04733605680915742926209147663, 8.386149095160429968169633445221, 9.871430220145332561055245097390, 11.92947475079383241193909707056, 12.35425705758004416410020428662, 13.37270266099706413270908646421, 15.48468581668417735062599004778
