| L(s) = 1 | + (−1.73 + 0.369i)2-s + (0.0646 + 0.614i)3-s + (1.05 − 0.471i)4-s + (1.85 − 2.06i)5-s + (−0.339 − 1.04i)6-s + (1.20 − 0.878i)8-s + (2.56 − 0.544i)9-s + (−2.46 + 4.27i)10-s + (−3.26 − 0.561i)11-s + (0.358 + 0.620i)12-s + (1.32 − 4.08i)13-s + (1.38 + 1.00i)15-s + (−3.32 + 3.69i)16-s + (−2.69 − 0.572i)17-s + (−4.25 + 1.89i)18-s + (−1.77 − 0.788i)19-s + ⋯ |
| L(s) = 1 | + (−1.22 + 0.261i)2-s + (0.0372 + 0.354i)3-s + (0.529 − 0.235i)4-s + (0.831 − 0.923i)5-s + (−0.138 − 0.426i)6-s + (0.427 − 0.310i)8-s + (0.853 − 0.181i)9-s + (−0.780 + 1.35i)10-s + (−0.985 − 0.169i)11-s + (0.103 + 0.179i)12-s + (0.367 − 1.13i)13-s + (0.358 + 0.260i)15-s + (−0.832 + 0.924i)16-s + (−0.653 − 0.138i)17-s + (−1.00 + 0.446i)18-s + (−0.406 − 0.180i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.632357 - 0.392404i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.632357 - 0.392404i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 + (3.26 + 0.561i)T \) |
| good | 2 | \( 1 + (1.73 - 0.369i)T + (1.82 - 0.813i)T^{2} \) |
| 3 | \( 1 + (-0.0646 - 0.614i)T + (-2.93 + 0.623i)T^{2} \) |
| 5 | \( 1 + (-1.85 + 2.06i)T + (-0.522 - 4.97i)T^{2} \) |
| 13 | \( 1 + (-1.32 + 4.08i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (2.69 + 0.572i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (1.77 + 0.788i)T + (12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (2.18 + 3.78i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (6.98 + 5.07i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.134 - 0.149i)T + (-3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (0.108 - 1.02i)T + (-36.1 - 7.69i)T^{2} \) |
| 41 | \( 1 + (-7.77 + 5.64i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 4.70T + 43T^{2} \) |
| 47 | \( 1 + (-11.9 - 5.30i)T + (31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (2.60 + 2.89i)T + (-5.54 + 52.7i)T^{2} \) |
| 59 | \( 1 + (-7.81 + 3.48i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (-0.661 + 0.734i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (-2.70 + 4.69i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.623 + 1.91i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-9.10 + 4.05i)T + (48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (6.15 - 1.30i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (-0.531 - 1.63i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (7.65 + 13.2i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (3.58 - 11.0i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.36630178761796967020558536706, −9.658686575791286324201837895873, −8.985685014505168032655359546220, −8.199912443503448481807647206520, −7.39443762851927810161668125957, −6.14005594554703071420738598122, −5.11391176626270875772556610768, −4.05771167234569976432308862758, −2.14206530850284289128481118714, −0.65691979087499963968337557404,
1.67687613678664592924335244339, 2.39693178823830380205299030729, 4.22625710143724851365697723231, 5.63353909627410399364773377022, 6.84638296119589504140347081122, 7.38145450338439051424066819918, 8.427345981140122192847632309946, 9.411004584842074821736154411726, 10.02542699814066304223407717790, 10.75083208596312847166637629105
