| L(s) = 1 | + (1.47 + 3.05i)2-s + (−0.584 − 0.856i)3-s + (−4.67 + 5.86i)4-s + (−0.262 + 0.283i)5-s + (1.75 − 3.04i)6-s + (4.87 − 2.81i)7-s + (−11.5 − 2.64i)8-s + (2.89 − 7.37i)9-s + (−1.25 − 0.386i)10-s + (−0.00121 − 0.00152i)11-s + (7.75 + 0.581i)12-s + (−15.9 + 4.90i)13-s + (15.7 + 10.7i)14-s + (0.396 + 0.0597i)15-s + (−2.28 − 10.0i)16-s + (23.3 − 21.6i)17-s + ⋯ |
| L(s) = 1 | + (0.735 + 1.52i)2-s + (−0.194 − 0.285i)3-s + (−1.16 + 1.46i)4-s + (−0.0525 + 0.0566i)5-s + (0.293 − 0.507i)6-s + (0.696 − 0.402i)7-s + (−1.44 − 0.330i)8-s + (0.321 − 0.819i)9-s + (−0.125 − 0.0386i)10-s + (−0.000110 − 0.000138i)11-s + (0.646 + 0.0484i)12-s + (−1.22 + 0.377i)13-s + (1.12 + 0.768i)14-s + (0.0264 + 0.00398i)15-s + (−0.142 − 0.625i)16-s + (1.37 − 1.27i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0902 - 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0902 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.946541 + 1.03618i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.946541 + 1.03618i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-10.6 - 41.6i)T \) |
| good | 2 | \( 1 + (-1.47 - 3.05i)T + (-2.49 + 3.12i)T^{2} \) |
| 3 | \( 1 + (0.584 + 0.856i)T + (-3.28 + 8.37i)T^{2} \) |
| 5 | \( 1 + (0.262 - 0.283i)T + (-1.86 - 24.9i)T^{2} \) |
| 7 | \( 1 + (-4.87 + 2.81i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (0.00121 + 0.00152i)T + (-26.9 + 117. i)T^{2} \) |
| 13 | \( 1 + (15.9 - 4.90i)T + (139. - 95.2i)T^{2} \) |
| 17 | \( 1 + (-23.3 + 21.6i)T + (21.5 - 288. i)T^{2} \) |
| 19 | \( 1 + (18.8 - 7.40i)T + (264. - 245. i)T^{2} \) |
| 23 | \( 1 + (31.7 - 4.77i)T + (505. - 155. i)T^{2} \) |
| 29 | \( 1 + (-2.19 + 3.22i)T + (-307. - 782. i)T^{2} \) |
| 31 | \( 1 + (-1.68 + 22.4i)T + (-950. - 143. i)T^{2} \) |
| 37 | \( 1 + (-0.370 - 0.213i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-44.3 + 21.3i)T + (1.04e3 - 1.31e3i)T^{2} \) |
| 47 | \( 1 + (-0.177 + 0.222i)T + (-491. - 2.15e3i)T^{2} \) |
| 53 | \( 1 + (-40.4 - 12.4i)T + (2.32e3 + 1.58e3i)T^{2} \) |
| 59 | \( 1 + (12.9 + 56.5i)T + (-3.13e3 + 1.51e3i)T^{2} \) |
| 61 | \( 1 + (-98.2 + 7.36i)T + (3.67e3 - 554. i)T^{2} \) |
| 67 | \( 1 + (-21.5 - 54.9i)T + (-3.29e3 + 3.05e3i)T^{2} \) |
| 71 | \( 1 + (13.8 - 91.9i)T + (-4.81e3 - 1.48e3i)T^{2} \) |
| 73 | \( 1 + (-10.1 - 32.9i)T + (-4.40e3 + 3.00e3i)T^{2} \) |
| 79 | \( 1 + (39.6 + 68.7i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-20.6 + 14.0i)T + (2.51e3 - 6.41e3i)T^{2} \) |
| 89 | \( 1 + (62.4 + 91.6i)T + (-2.89e3 + 7.37e3i)T^{2} \) |
| 97 | \( 1 + (-4.24 - 5.31i)T + (-2.09e3 + 9.17e3i)T^{2} \) |
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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
−15.95325754120287355125645318656, −14.67389353061377226361843967235, −14.25063338822218309009837042532, −12.81542838844802037016626916969, −11.76611258919869141016387170836, −9.666579853591466407617986298948, −7.86489066189553614867628487127, −7.06505377075898284313865389291, −5.63606780229898146945644066122, −4.20643609318340487234498744325,
2.18982643649290914202616227467, 4.25906116609162595176925686285, 5.42570425273048505196441468387, 8.098794912246220547451284044914, 10.03884029457649480558670505733, 10.66641638949726090072128319092, 12.02568954618620476834357071697, 12.67799479331380369753138811102, 14.10570433810005418104993319948, 14.95470318622710163162899990569
