| L(s) = 1 | + (−1.62 + 2.03i)2-s + (−6.93 + 1.04i)3-s + (0.276 + 1.21i)4-s + (6.69 − 4.56i)5-s + (9.11 − 15.7i)6-s + (−16.5 − 28.6i)7-s + (−21.6 − 10.4i)8-s + (21.1 − 6.52i)9-s + (−1.57 + 21.0i)10-s + (0.540 − 2.36i)11-s + (−3.18 − 8.11i)12-s + (4.87 + 64.9i)13-s + (85.0 + 12.8i)14-s + (−41.6 + 38.6i)15-s + (47.3 − 22.7i)16-s + (−78.9 − 53.8i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 + 0.718i)2-s + (−1.33 + 0.201i)3-s + (0.0345 + 0.151i)4-s + (0.599 − 0.408i)5-s + (0.619 − 1.07i)6-s + (−0.893 − 1.54i)7-s + (−0.956 − 0.460i)8-s + (0.783 − 0.241i)9-s + (−0.0497 + 0.664i)10-s + (0.0148 − 0.0648i)11-s + (−0.0766 − 0.195i)12-s + (0.103 + 1.38i)13-s + (1.62 + 0.244i)14-s + (−0.717 + 0.665i)15-s + (0.739 − 0.355i)16-s + (−1.12 − 0.768i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.330 + 0.943i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.330 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0703233 - 0.0991030i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0703233 - 0.0991030i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-259. - 111. i)T \) |
| good | 2 | \( 1 + (1.62 - 2.03i)T + (-1.78 - 7.79i)T^{2} \) |
| 3 | \( 1 + (6.93 - 1.04i)T + (25.8 - 7.95i)T^{2} \) |
| 5 | \( 1 + (-6.69 + 4.56i)T + (45.6 - 116. i)T^{2} \) |
| 7 | \( 1 + (16.5 + 28.6i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-0.540 + 2.36i)T + (-1.19e3 - 577. i)T^{2} \) |
| 13 | \( 1 + (-4.87 - 64.9i)T + (-2.17e3 + 327. i)T^{2} \) |
| 17 | \( 1 + (78.9 + 53.8i)T + (1.79e3 + 4.57e3i)T^{2} \) |
| 19 | \( 1 + (98.5 + 30.3i)T + (5.66e3 + 3.86e3i)T^{2} \) |
| 23 | \( 1 + (71.4 + 66.2i)T + (909. + 1.21e4i)T^{2} \) |
| 29 | \( 1 + (-5.06 - 0.764i)T + (2.33e4 + 7.18e3i)T^{2} \) |
| 31 | \( 1 + (-41.3 - 105. i)T + (-2.18e4 + 2.02e4i)T^{2} \) |
| 37 | \( 1 + (145. - 252. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-136. + 170. i)T + (-1.53e4 - 6.71e4i)T^{2} \) |
| 47 | \( 1 + (26.9 + 117. i)T + (-9.35e4 + 4.50e4i)T^{2} \) |
| 53 | \( 1 + (-25.3 + 338. i)T + (-1.47e5 - 2.21e4i)T^{2} \) |
| 59 | \( 1 + (17.3 - 8.36i)T + (1.28e5 - 1.60e5i)T^{2} \) |
| 61 | \( 1 + (-184. + 470. i)T + (-1.66e5 - 1.54e5i)T^{2} \) |
| 67 | \( 1 + (371. + 114. i)T + (2.48e5 + 1.69e5i)T^{2} \) |
| 71 | \( 1 + (638. - 592. i)T + (2.67e4 - 3.56e5i)T^{2} \) |
| 73 | \( 1 + (34.5 + 460. i)T + (-3.84e5 + 5.79e4i)T^{2} \) |
| 79 | \( 1 + (-7.37 - 12.7i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (712. - 107. i)T + (5.46e5 - 1.68e5i)T^{2} \) |
| 89 | \( 1 + (-481. + 72.6i)T + (6.73e5 - 2.07e5i)T^{2} \) |
| 97 | \( 1 + (-89.0 + 390. i)T + (-8.22e5 - 3.95e5i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.89774355537251328438325845474, −13.84967065016714307612006019850, −12.75183347154123259889231466369, −11.39083987910296208326624416190, −10.16254138146655422344733312549, −8.966178789975260251132246977134, −6.92962372078708824529366744169, −6.37295680926903238486974589541, −4.37617489244562917026733277617, −0.12792910419528377439181452888,
2.38237878435077612572062412341, 5.92249076769582353165512436198, 6.03710686254329825504459777407, 8.793073301098294581691332159721, 10.11066371249099885304971744287, 10.90154758641991585755969989687, 12.11561178115531101832510012273, 12.87058456288191838174574664095, 14.96009962529338721768550644989, 15.84330812421954785688594828972
