| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 - 0.381i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.924 - 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.56334 + 0.705462i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.56334 + 0.705462i\) |
| \(L(\frac{3}{2})\) |
|
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
−10.91056865429620774810077634161, −10.37144219863921597161490738215, −9.516126856139168119080237890730, −7.944615627972618855784344124905, −7.01198170094798215027215947084, −5.86139853708990879109377143643, −5.02527605136658912414155838133, −4.02198462996491978219530622525, −3.49638040236676720766889862633, −2.17696379782492403167956019536,
1.63457770107811088461772359975, 3.20828827327577815918043991698, 4.07294419510889762241386536752, 5.21017705217359397630946843266, 6.02462949628862744390171782200, 6.95641306708217321395487867406, 7.70794654050289283148896897373, 8.604069163426869322723433532042, 10.06154115394621706829714068373, 11.38683678719313593693073093884
