| L(s) = 1 | + (0.102 + 0.315i)2-s + (1.52 − 1.11i)4-s + (0.962 − 1.66i)5-s + (−1.29 − 0.575i)7-s + (1.04 + 0.757i)8-s + (0.624 + 0.132i)10-s + (−0.138 − 1.32i)11-s + (−0.607 − 0.674i)13-s + (0.0489 − 0.466i)14-s + (1.03 − 3.18i)16-s + (0.279 − 2.65i)17-s + (−4.34 + 4.82i)19-s + (−0.380 − 3.62i)20-s + (0.401 − 0.178i)22-s + (6.78 + 4.92i)23-s + ⋯ |
| L(s) = 1 | + (0.0724 + 0.222i)2-s + (0.764 − 0.555i)4-s + (0.430 − 0.745i)5-s + (−0.488 − 0.217i)7-s + (0.368 + 0.267i)8-s + (0.197 + 0.0419i)10-s + (−0.0418 − 0.398i)11-s + (−0.168 − 0.186i)13-s + (0.0130 − 0.124i)14-s + (0.259 − 0.797i)16-s + (0.0677 − 0.644i)17-s + (−0.996 + 1.10i)19-s + (−0.0850 − 0.809i)20-s + (0.0856 − 0.0381i)22-s + (1.41 + 1.02i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 279 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.804 + 0.594i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 279 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.804 + 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.50296 - 0.494875i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.50296 - 0.494875i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 31 | \( 1 + (-5.43 - 1.21i)T \) |
| good | 2 | \( 1 + (-0.102 - 0.315i)T + (-1.61 + 1.17i)T^{2} \) |
| 5 | \( 1 + (-0.962 + 1.66i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (1.29 + 0.575i)T + (4.68 + 5.20i)T^{2} \) |
| 11 | \( 1 + (0.138 + 1.32i)T + (-10.7 + 2.28i)T^{2} \) |
| 13 | \( 1 + (0.607 + 0.674i)T + (-1.35 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.279 + 2.65i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (4.34 - 4.82i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (-6.78 - 4.92i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.154 - 0.476i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (-0.749 - 1.29i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (6.14 + 1.30i)T + (37.4 + 16.6i)T^{2} \) |
| 43 | \( 1 + (-2.70 + 3.00i)T + (-4.49 - 42.7i)T^{2} \) |
| 47 | \( 1 + (4.08 - 12.5i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (3.23 - 1.43i)T + (35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (6.73 - 1.43i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + 2.87T + 61T^{2} \) |
| 67 | \( 1 + (3.11 - 5.39i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (5.94 - 2.64i)T + (47.5 - 52.7i)T^{2} \) |
| 73 | \( 1 + (-1.44 - 13.7i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (-1.24 + 11.8i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (-11.7 - 2.49i)T + (75.8 + 33.7i)T^{2} \) |
| 89 | \( 1 + (-11.9 + 8.66i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-6.74 + 4.90i)T + (29.9 - 92.2i)T^{2} \) |
| 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
−11.74861609667677908478878237043, −10.76746537594758774555557396332, −9.917835494197452444680973131919, −9.008921806606632213973607488625, −7.78357201410711194325753753882, −6.70242867133609247935329507135, −5.77672618230003581154133827466, −4.84175837853479566776212164491, −3.05740629421152155143555498498, −1.37671052194481078027871440841,
2.24245767480336028118667795798, 3.13066809393411927536350660122, 4.64089224188886349711473434734, 6.50822283170499902972120142769, 6.66820860806304268237754067683, 8.036020659671750381583233980755, 9.170904873722579884576780784504, 10.40505964758735187147506208307, 10.87839451441603274851855690516, 11.99510229838220806666191656640
