| L(s) = 1 | + (−0.142 + 0.989i)2-s + (−0.390 − 0.450i)3-s + (−0.959 − 0.281i)4-s + (−0.841 − 0.540i)5-s + (0.501 − 0.322i)6-s + (−0.293 + 2.04i)7-s + (0.415 − 0.909i)8-s + (0.376 − 2.61i)9-s + (0.654 − 0.755i)10-s + (−0.807 − 0.519i)11-s + (0.247 + 0.541i)12-s + (2.47 + 5.41i)13-s + (−1.97 − 0.580i)14-s + (0.0847 + 0.589i)15-s + (0.841 + 0.540i)16-s + (−5.08 + 1.49i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 + 0.699i)2-s + (−0.225 − 0.259i)3-s + (−0.479 − 0.140i)4-s + (−0.376 − 0.241i)5-s + (0.204 − 0.131i)6-s + (−0.110 + 0.771i)7-s + (0.146 − 0.321i)8-s + (0.125 − 0.872i)9-s + (0.207 − 0.238i)10-s + (−0.243 − 0.156i)11-s + (0.0714 + 0.156i)12-s + (0.686 + 1.50i)13-s + (−0.528 − 0.155i)14-s + (0.0218 + 0.152i)15-s + (0.210 + 0.135i)16-s + (−1.23 + 0.362i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.288013 + 0.703503i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.288013 + 0.703503i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 - 0.989i)T \) |
| 5 | \( 1 + (0.841 + 0.540i)T \) |
| 67 | \( 1 + (1.85 - 7.97i)T \) |
| good | 3 | \( 1 + (0.390 + 0.450i)T + (-0.426 + 2.96i)T^{2} \) |
| 7 | \( 1 + (0.293 - 2.04i)T + (-6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (0.807 + 0.519i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-2.47 - 5.41i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (5.08 - 1.49i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-0.218 - 1.52i)T + (-18.2 + 5.35i)T^{2} \) |
| 23 | \( 1 + (-3.21 - 3.71i)T + (-3.27 + 22.7i)T^{2} \) |
| 29 | \( 1 + 0.900T + 29T^{2} \) |
| 31 | \( 1 + (3.67 - 8.04i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 - 4.68T + 37T^{2} \) |
| 41 | \( 1 + (7.62 - 2.24i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (7.88 - 2.31i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + (-0.484 - 0.559i)T + (-6.68 + 46.5i)T^{2} \) |
| 53 | \( 1 + (-0.126 - 0.0372i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (0.902 - 1.97i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (4.69 - 3.01i)T + (25.3 - 55.4i)T^{2} \) |
| 71 | \( 1 + (-12.6 - 3.72i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-12.2 + 7.84i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (2.89 + 6.33i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-0.517 - 0.332i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (6.97 - 8.05i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
| 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
−11.00438085058642322413170695768, −9.603403545846737234314912548859, −8.932747795131806635042667593096, −8.405010978863393596906213249902, −7.05571310420594920381145755325, −6.53496581154824361595226999506, −5.61954235752811580955599844074, −4.50165680232932321182246069714, −3.45211944710124112030842342384, −1.58431704149938084550077636338,
0.44238551648046719142013316876, 2.31615282406176542368057779965, 3.49741309561917450586578660554, 4.49541230182801035178569072510, 5.36413753298187244604952179304, 6.73490960206309003011839761113, 7.73087833248321063195155288906, 8.418563263208075091803580288795, 9.581957685815594285361196852268, 10.52650025089061685488977712965
