L(s) = 1 | − 2-s + (0.5 − 0.866i)3-s + 4-s + (−0.769 + 1.33i)5-s + (−0.5 + 0.866i)6-s + (−2.22 + 1.43i)7-s − 8-s + (−0.499 − 0.866i)9-s + (0.769 − 1.33i)10-s + (3.19 − 5.52i)11-s + (0.5 − 0.866i)12-s + (0.520 + 3.56i)13-s + (2.22 − 1.43i)14-s + (0.769 + 1.33i)15-s + 16-s + 4.38·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.288 − 0.499i)3-s + 0.5·4-s + (−0.344 + 0.596i)5-s + (−0.204 + 0.353i)6-s + (−0.840 + 0.542i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.243 − 0.421i)10-s + (0.962 − 1.66i)11-s + (0.144 − 0.249i)12-s + (0.144 + 0.989i)13-s + (0.593 − 0.383i)14-s + (0.198 + 0.344i)15-s + 0.250·16-s + 1.06·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00263i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00263i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.07790 + 0.00142222i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.07790 + 0.00142222i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (2.22 - 1.43i)T \) |
| 13 | \( 1 + (-0.520 - 3.56i)T \) |
good | 5 | \( 1 + (0.769 - 1.33i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.19 + 5.52i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 - 4.38T + 17T^{2} \) |
| 19 | \( 1 + (0.0509 + 0.0882i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 9.09T + 23T^{2} \) |
| 29 | \( 1 + (-3.51 - 6.08i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.611 + 1.05i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 3.84T + 37T^{2} \) |
| 41 | \( 1 + (2.42 + 4.19i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.877 - 1.51i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (2.07 - 3.58i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.11 + 7.12i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 7.13T + 59T^{2} \) |
| 61 | \( 1 + (-3.08 - 5.34i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.857 + 1.48i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.57 + 7.92i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.82 - 8.35i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.03 - 3.52i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 7.02T + 83T^{2} \) |
| 89 | \( 1 + 15.5T + 89T^{2} \) |
| 97 | \( 1 + (-0.996 + 1.72i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.99112152488002518238349682120, −9.669107465671870559985848652672, −8.946604392399250770613558536646, −8.369165163796799744336735410652, −7.01599652568382591126747409707, −6.63866323735595280878010462400, −5.58796301512270883896993001749, −3.51734602380226913905034998266, −2.94136081144713257299257382002, −1.13469009728378678193571582098,
1.02575454851706550483919886889, 2.89035846000635803807751267572, 4.02772741475735250797997860785, 5.05746556255090099428429540674, 6.49789798027842061757186793419, 7.35821441906831794683488948167, 8.223396600695188289200872906923, 9.225946086463712809548696468728, 9.850196188523277638596784413360, 10.40944490563945253379069139977