| L(s) = 1 | + (−0.173 − 0.984i)2-s + (1.31 − 1.10i)3-s + (−0.939 + 0.342i)4-s + (−3.37 − 1.22i)5-s + (−1.31 − 1.10i)6-s + (−1.75 + 3.03i)7-s + (0.5 + 0.866i)8-s + (−0.00647 + 0.0367i)9-s + (−0.623 + 3.53i)10-s + (−0.5 − 0.866i)11-s + (−0.860 + 1.49i)12-s + (−4.44 − 3.72i)13-s + (3.29 + 1.19i)14-s + (−5.81 + 2.11i)15-s + (0.766 − 0.642i)16-s + (0.138 + 0.787i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 − 0.696i)2-s + (0.761 − 0.638i)3-s + (−0.469 + 0.171i)4-s + (−1.51 − 0.549i)5-s + (−0.538 − 0.451i)6-s + (−0.661 + 1.14i)7-s + (0.176 + 0.306i)8-s + (−0.00215 + 0.0122i)9-s + (−0.197 + 1.11i)10-s + (−0.150 − 0.261i)11-s + (−0.248 + 0.430i)12-s + (−1.23 − 1.03i)13-s + (0.879 + 0.320i)14-s + (−1.50 + 0.546i)15-s + (0.191 − 0.160i)16-s + (0.0336 + 0.191i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0852649 + 0.170643i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0852649 + 0.170643i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.173 + 0.984i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (4.33 - 0.455i)T \) |
| good | 3 | \( 1 + (-1.31 + 1.10i)T + (0.520 - 2.95i)T^{2} \) |
| 5 | \( 1 + (3.37 + 1.22i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (1.75 - 3.03i)T + (-3.5 - 6.06i)T^{2} \) |
| 13 | \( 1 + (4.44 + 3.72i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.138 - 0.787i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (2.27 - 0.828i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.21 + 6.89i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-0.937 + 1.62i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 3.30T + 37T^{2} \) |
| 41 | \( 1 + (-7.35 + 6.16i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (7.07 + 2.57i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (0.742 - 4.21i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-5.86 + 2.13i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.222 - 1.26i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (12.1 - 4.42i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-2.71 + 15.4i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (0.213 + 0.0778i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (7.09 - 5.95i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (0.701 - 0.588i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (5.06 - 8.76i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (12.5 + 10.5i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-1.92 - 10.9i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.70893994018891542065584436144, −9.594558659789087169374872146857, −8.542142380970995871993292127292, −8.148646008545692044190241192665, −7.31830685329106521160465078910, −5.68279139904656175520507212778, −4.42571706206009692083317877184, −3.17399342769802961399921228270, −2.30575133636661107553887825479, −0.11061470685213029877188875853,
3.03642460181063566092227612906, 4.05706205388872019177227006193, 4.56601241810951774810236077452, 6.68824971796728521762015676857, 7.12094009858072468210773885122, 8.032911115082583600563760061056, 8.966658936577721667432008674644, 9.923846981712474920342173380900, 10.58816499879630212564094082656, 11.77106764458811239720696284318
