| L(s) = 1 | + (0.417 − 2.36i)2-s + (−3.54 − 1.28i)4-s + (−0.0713 + 0.0598i)5-s + (0.544 − 0.198i)7-s + (−2.12 + 3.67i)8-s + (0.111 + 0.193i)10-s + (2.36 + 1.98i)11-s + (0.729 + 4.13i)13-s + (−0.241 − 1.37i)14-s + (2.03 + 1.71i)16-s + (−0.995 − 1.72i)17-s + (1.92 − 3.33i)19-s + (0.329 − 0.119i)20-s + (5.69 − 4.77i)22-s + (−4.18 − 1.52i)23-s + ⋯ |
| L(s) = 1 | + (0.294 − 1.67i)2-s + (−1.77 − 0.644i)4-s + (−0.0318 + 0.0267i)5-s + (0.205 − 0.0749i)7-s + (−0.750 + 1.29i)8-s + (0.0353 + 0.0612i)10-s + (0.714 + 0.599i)11-s + (0.202 + 1.14i)13-s + (−0.0646 − 0.366i)14-s + (0.509 + 0.427i)16-s + (−0.241 − 0.418i)17-s + (0.441 − 0.764i)19-s + (0.0736 − 0.0268i)20-s + (1.21 − 1.01i)22-s + (−0.872 − 0.317i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.490 + 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.490 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.522028 - 0.893362i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.522028 - 0.893362i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (-0.417 + 2.36i)T + (-1.87 - 0.684i)T^{2} \) |
| 5 | \( 1 + (0.0713 - 0.0598i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (-0.544 + 0.198i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (-2.36 - 1.98i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.729 - 4.13i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (0.995 + 1.72i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.92 + 3.33i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.18 + 1.52i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (1.11 - 6.30i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (1.55 + 0.566i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (2.01 + 3.49i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.190 - 1.07i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (5.28 + 4.43i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-3.37 + 1.22i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + 5.40T + 53T^{2} \) |
| 59 | \( 1 + (-7.87 + 6.61i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-12.4 + 4.51i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (1.53 + 8.70i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-0.572 - 0.991i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.0977 - 0.169i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.25 - 7.09i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-2.58 + 14.6i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (0.776 - 1.34i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.05 - 3.40i)T + (16.8 + 95.5i)T^{2} \) |
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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
−13.78039599186287460948652984720, −12.68605482314522015223801863168, −11.70385704205471109853982682203, −11.06786410921081926809085955749, −9.734145238010580559322598672326, −8.964226241834396725485239788233, −6.97232819643360560934389184074, −4.89889003732022284234093291981, −3.69122795591946895028701248533, −1.86540728154967913402049276079,
3.93545098214888768680473724978, 5.52058241346123671325103760335, 6.38836110722858213220528364097, 7.84801545511999548149200677099, 8.526836418353905960311181751188, 10.02884783058399261150168077370, 11.67959131168413229420835973769, 13.04518032501323883228659768794, 13.98941219242717048836570358849, 14.80888014054293749977325951973
