| L(s) = 1 | + (−0.0879 + 0.611i)2-s + (−0.928 − 2.03i)3-s + (1.55 + 0.455i)4-s + (−0.809 − 0.934i)5-s + (1.32 − 0.389i)6-s + (2.72 + 1.74i)7-s + (−0.928 + 2.03i)8-s + (−1.30 + 1.51i)9-s + (0.642 − 0.413i)10-s + (0.745 + 5.18i)11-s + (−0.514 − 3.58i)12-s + (2.52 − 1.62i)13-s + (−1.30 + 1.51i)14-s + (−1.14 + 2.51i)15-s + (1.55 + 1.00i)16-s + (−0.732 + 0.215i)17-s + ⋯ |
| L(s) = 1 | + (−0.0621 + 0.432i)2-s + (−0.536 − 1.17i)3-s + (0.776 + 0.227i)4-s + (−0.361 − 0.417i)5-s + (0.541 − 0.158i)6-s + (1.02 + 0.661i)7-s + (−0.328 + 0.719i)8-s + (−0.436 + 0.503i)9-s + (0.203 − 0.130i)10-s + (0.224 + 1.56i)11-s + (−0.148 − 1.03i)12-s + (0.699 − 0.449i)13-s + (−0.350 + 0.403i)14-s + (−0.296 + 0.649i)15-s + (0.389 + 0.250i)16-s + (−0.177 + 0.0521i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0235i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.54252 + 0.0181663i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.54252 + 0.0181663i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 \) |
| good | 2 | \( 1 + (0.0879 - 0.611i)T + (-1.91 - 0.563i)T^{2} \) |
| 3 | \( 1 + (0.928 + 2.03i)T + (-1.96 + 2.26i)T^{2} \) |
| 5 | \( 1 + (0.809 + 0.934i)T + (-0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (-2.72 - 1.74i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.745 - 5.18i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-2.52 + 1.62i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (0.732 - 0.215i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-1.91 - 0.563i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-2.87 + 0.845i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-2.78 + 6.10i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-0.809 + 0.934i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-2.27 - 2.62i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + 2.23T + 47T^{2} \) |
| 53 | \( 1 + (-0.397 - 0.255i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-5.44 + 3.49i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (2.88 - 6.31i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-0.393 + 2.73i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (1.74 - 12.1i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (6.26 + 1.83i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (9.20 - 5.91i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-5.73 + 6.62i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (4.35 + 9.52i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (11.5 + 13.3i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−11.36606156821493092730826378111, −10.03983814844553311690646717224, −8.612760282753402758975995736590, −7.922000449893984113448453515176, −7.29189721788477901788471867004, −6.38713303472371425373482032085, −5.56165998687766941446932018267, −4.39625017102476894134995940169, −2.41454698462316388713017298989, −1.41967484067096512604098432793,
1.23553966420902331056377129713, 3.15981391568301781727496630513, 3.97388270589589738645101440450, 5.12479483728817373458808887561, 6.14040100817080446885490876849, 7.14105531569238978102241368814, 8.268588490046942607977713074299, 9.313849120537569906323547758437, 10.45693843830519317270680506519, 10.90349568282905120828805254958
