| L(s) = 1 | + (3.27 − 4.10i)2-s + (−9.59 + 1.44i)3-s + (−4.35 − 19.0i)4-s + (4.62 − 3.15i)5-s + (−25.4 + 44.0i)6-s + (0.410 + 0.710i)7-s + (−54.6 − 26.3i)8-s + (64.0 − 19.7i)9-s + (2.19 − 29.3i)10-s + (10.7 − 47.2i)11-s + (69.3 + 176. i)12-s + (0.341 + 4.55i)13-s + (4.25 + 0.641i)14-s + (−39.8 + 36.9i)15-s + (−146. + 70.3i)16-s + (89.5 + 61.0i)17-s + ⋯ |
| L(s) = 1 | + (1.15 − 1.45i)2-s + (−1.84 + 0.278i)3-s + (−0.544 − 2.38i)4-s + (0.413 − 0.282i)5-s + (−1.73 + 3.00i)6-s + (0.0221 + 0.0383i)7-s + (−2.41 − 1.16i)8-s + (2.37 − 0.732i)9-s + (0.0694 − 0.926i)10-s + (0.295 − 1.29i)11-s + (1.66 + 4.24i)12-s + (0.00728 + 0.0972i)13-s + (0.0812 + 0.0122i)14-s + (−0.685 + 0.635i)15-s + (−2.28 + 1.09i)16-s + (1.27 + 0.871i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.916 + 0.399i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.916 + 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.273684 - 1.31350i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.273684 - 1.31350i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-94.5 - 265. i)T \) |
| good | 2 | \( 1 + (-3.27 + 4.10i)T + (-1.78 - 7.79i)T^{2} \) |
| 3 | \( 1 + (9.59 - 1.44i)T + (25.8 - 7.95i)T^{2} \) |
| 5 | \( 1 + (-4.62 + 3.15i)T + (45.6 - 116. i)T^{2} \) |
| 7 | \( 1 + (-0.410 - 0.710i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-10.7 + 47.2i)T + (-1.19e3 - 577. i)T^{2} \) |
| 13 | \( 1 + (-0.341 - 4.55i)T + (-2.17e3 + 327. i)T^{2} \) |
| 17 | \( 1 + (-89.5 - 61.0i)T + (1.79e3 + 4.57e3i)T^{2} \) |
| 19 | \( 1 + (23.1 + 7.12i)T + (5.66e3 + 3.86e3i)T^{2} \) |
| 23 | \( 1 + (-0.872 - 0.809i)T + (909. + 1.21e4i)T^{2} \) |
| 29 | \( 1 + (-31.9 - 4.80i)T + (2.33e4 + 7.18e3i)T^{2} \) |
| 31 | \( 1 + (21.6 + 55.0i)T + (-2.18e4 + 2.02e4i)T^{2} \) |
| 37 | \( 1 + (-176. + 306. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (39.1 - 49.1i)T + (-1.53e4 - 6.71e4i)T^{2} \) |
| 47 | \( 1 + (-47.4 - 207. i)T + (-9.35e4 + 4.50e4i)T^{2} \) |
| 53 | \( 1 + (8.92 - 119. i)T + (-1.47e5 - 2.21e4i)T^{2} \) |
| 59 | \( 1 + (-330. + 159. i)T + (1.28e5 - 1.60e5i)T^{2} \) |
| 61 | \( 1 + (-13.4 + 34.2i)T + (-1.66e5 - 1.54e5i)T^{2} \) |
| 67 | \( 1 + (-165. - 51.1i)T + (2.48e5 + 1.69e5i)T^{2} \) |
| 71 | \( 1 + (673. - 624. i)T + (2.67e4 - 3.56e5i)T^{2} \) |
| 73 | \( 1 + (-2.53 - 33.7i)T + (-3.84e5 + 5.79e4i)T^{2} \) |
| 79 | \( 1 + (-306. - 530. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-411. + 62.0i)T + (5.46e5 - 1.68e5i)T^{2} \) |
| 89 | \( 1 + (911. - 137. i)T + (6.73e5 - 2.07e5i)T^{2} \) |
| 97 | \( 1 + (96.1 - 421. i)T + (-8.22e5 - 3.95e5i)T^{2} \) |
| 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
−14.64795133048332434696934269749, −13.24068887379156646344666325290, −12.39254559370178178009722871609, −11.42663502413946903585042970967, −10.73316730316765277213065407837, −9.656438289934633810119741533844, −6.04978601218645780075815083049, −5.43154643192434207050081078686, −3.93481368639448211856374884345, −1.06467142440298260491877641253,
4.54237655740066081904099571010, 5.58042938366954707334120762664, 6.61813636974391116760670183104, 7.49099808198351820411227784249, 10.08451039792161076962522821907, 11.89115384337053517266814562455, 12.47345255533770382040522261771, 13.69157289403210454955715753370, 14.96400729868217234674654939885, 16.07212567052564448552879093336
