| L(s) = 1 | + (1.01 − 0.453i)2-s + (2.79 − 0.594i)3-s + (−0.507 + 0.563i)4-s + (−0.361 − 3.43i)5-s + (2.57 − 1.87i)6-s + (−0.949 + 2.92i)8-s + (4.72 − 2.10i)9-s + (−1.92 − 3.33i)10-s + (−1.73 − 2.82i)11-s + (−1.08 + 1.87i)12-s + (1.66 + 1.21i)13-s + (−3.05 − 9.39i)15-s + (0.199 + 1.89i)16-s + (1.76 + 0.786i)17-s + (3.85 − 4.28i)18-s + (1.08 + 1.20i)19-s + ⋯ |
| L(s) = 1 | + (0.719 − 0.320i)2-s + (1.61 − 0.343i)3-s + (−0.253 + 0.281i)4-s + (−0.161 − 1.53i)5-s + (1.05 − 0.764i)6-s + (−0.335 + 1.03i)8-s + (1.57 − 0.701i)9-s + (−0.609 − 1.05i)10-s + (−0.524 − 0.851i)11-s + (−0.312 + 0.541i)12-s + (0.462 + 0.335i)13-s + (−0.788 − 2.42i)15-s + (0.0499 + 0.474i)16-s + (0.428 + 0.190i)17-s + (0.909 − 1.00i)18-s + (0.249 + 0.276i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.434 + 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.54548 - 1.59918i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.54548 - 1.59918i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 + (1.73 + 2.82i)T \) |
| good | 2 | \( 1 + (-1.01 + 0.453i)T + (1.33 - 1.48i)T^{2} \) |
| 3 | \( 1 + (-2.79 + 0.594i)T + (2.74 - 1.22i)T^{2} \) |
| 5 | \( 1 + (0.361 + 3.43i)T + (-4.89 + 1.03i)T^{2} \) |
| 13 | \( 1 + (-1.66 - 1.21i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.76 - 0.786i)T + (11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (-1.08 - 1.20i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-0.403 + 0.698i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.46 - 7.58i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (0.0824 - 0.784i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (9.84 + 2.09i)T + (33.8 + 15.0i)T^{2} \) |
| 41 | \( 1 + (0.657 - 2.02i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 3.08T + 43T^{2} \) |
| 47 | \( 1 + (-5.06 - 5.62i)T + (-4.91 + 46.7i)T^{2} \) |
| 53 | \( 1 + (1.13 - 10.7i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (2.20 - 2.44i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (-0.112 - 1.07i)T + (-59.6 + 12.6i)T^{2} \) |
| 67 | \( 1 + (1.20 + 2.08i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.57 + 1.87i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.820 + 0.911i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (8.66 - 3.85i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (-13.0 + 9.44i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-2.21 + 3.84i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.23 + 3.80i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.72402021596506417862568077828, −9.268545353426572032617085421592, −8.736689171007785362545801721076, −8.306312545642284203612756818359, −7.46439866429727842726847552646, −5.70599349749362615730315093071, −4.70363479866339678011540348599, −3.73441634485665881750907891387, −2.93780703836598860031110024135, −1.46987557029695937646265969989,
2.33351583463261771433612308793, 3.30470539267455666991293133839, 3.99397644982320739243065340828, 5.23991182489403901328339713429, 6.57417833773367801279930262231, 7.34315068567510289170304342690, 8.177629617861781161780425714777, 9.346247178298681122217380667271, 10.06321823104917817735446963268, 10.58482253275037331837093672259
