| L(s) = 1 | + (2.37 + 0.505i)2-s + (−0.149 + 1.42i)3-s + (3.56 + 1.58i)4-s + (0.842 + 0.935i)5-s + (−1.07 + 3.31i)6-s + (3.74 + 2.72i)8-s + (0.922 + 0.196i)9-s + (1.52 + 2.64i)10-s + (−0.785 − 3.22i)11-s + (−2.80 + 4.85i)12-s + (−0.982 − 3.02i)13-s + (−1.46 + 1.06i)15-s + (2.30 + 2.56i)16-s + (−5.79 + 1.23i)17-s + (2.09 + 0.932i)18-s + (2.61 − 1.16i)19-s + ⋯ |
| L(s) = 1 | + (1.68 + 0.357i)2-s + (−0.0865 + 0.823i)3-s + (1.78 + 0.794i)4-s + (0.376 + 0.418i)5-s + (−0.439 + 1.35i)6-s + (1.32 + 0.963i)8-s + (0.307 + 0.0653i)9-s + (0.483 + 0.837i)10-s + (−0.236 − 0.971i)11-s + (−0.808 + 1.40i)12-s + (−0.272 − 0.838i)13-s + (−0.377 + 0.273i)15-s + (0.577 + 0.641i)16-s + (−1.40 + 0.298i)17-s + (0.493 + 0.219i)18-s + (0.599 − 0.267i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.245 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.245 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.92562 + 2.27820i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.92562 + 2.27820i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 + (0.785 + 3.22i)T \) |
| good | 2 | \( 1 + (-2.37 - 0.505i)T + (1.82 + 0.813i)T^{2} \) |
| 3 | \( 1 + (0.149 - 1.42i)T + (-2.93 - 0.623i)T^{2} \) |
| 5 | \( 1 + (-0.842 - 0.935i)T + (-0.522 + 4.97i)T^{2} \) |
| 13 | \( 1 + (0.982 + 3.02i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (5.79 - 1.23i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (-2.61 + 1.16i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (3.38 - 5.85i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.63 - 2.64i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-6.50 + 7.22i)T + (-3.24 - 30.8i)T^{2} \) |
| 37 | \( 1 + (-0.570 - 5.42i)T + (-36.1 + 7.69i)T^{2} \) |
| 41 | \( 1 + (-0.254 - 0.184i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 0.132T + 43T^{2} \) |
| 47 | \( 1 + (-8.56 + 3.81i)T + (31.4 - 34.9i)T^{2} \) |
| 53 | \( 1 + (-2.91 + 3.23i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (6.34 + 2.82i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (1.64 + 1.82i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (-4.70 - 8.15i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.0360 - 0.110i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.561 - 0.250i)T + (48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (8.34 + 1.77i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (-0.293 + 0.904i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (5.02 - 8.70i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.43 - 16.7i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−11.14912467148519456824017965537, −10.39044112599121001721430988073, −9.453422008109412072915922183577, −8.072713525747420207116597014384, −7.02178026850449413959434891080, −6.04408904207295839366014992675, −5.36341628744568759530553020189, −4.39787181713159534467298582732, −3.52750583227577092564461990917, −2.48962877995485154861806721222,
1.70877422104494742680149386223, 2.50812271383180465471521542218, 4.21012485775566251612936211363, 4.73943473225365231773695492553, 5.92446716065267616419414005373, 6.76217891770835426592948754231, 7.40950975853765113219310097736, 8.934409622942480754909534732528, 9.990640714800553182433405252209, 11.02043054482561642104733101016
