| 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 | \( 1 + (-6.74 - 4.90i)T + (29.9 + 92.2i)T^{2} \) |
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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
−11.99510229838220806666191656640, −10.87839451441603274851855690516, −10.40505964758735187147506208307, −9.170904873722579884576780784504, −8.036020659671750381583233980755, −6.66820860806304268237754067683, −6.50822283170499902972120142769, −4.64089224188886349711473434734, −3.13066809393411927536350660122, −2.24245767480336028118667795798,
1.37671052194481078027871440841, 3.05740629421152155143555498498, 4.84175837853479566776212164491, 5.77672618230003581154133827466, 6.70242867133609247935329507135, 7.78357201410711194325753753882, 9.008921806606632213973607488625, 9.917835494197452444680973131919, 10.76746537594758774555557396332, 11.74861609667677908478878237043
