L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s + 0.999·6-s + (−0.5 + 2.59i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (0.499 + 0.866i)10-s + (0.5 + 0.866i)11-s + (−0.499 + 0.866i)12-s − 13-s + (−2 − 1.73i)14-s − 0.999·15-s + (−0.5 + 0.866i)16-s + (3 + 5.19i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.223 − 0.387i)5-s + 0.408·6-s + (−0.188 + 0.981i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.158 + 0.273i)10-s + (0.150 + 0.261i)11-s + (−0.144 + 0.249i)12-s − 0.277·13-s + (−0.534 − 0.462i)14-s − 0.258·15-s + (−0.125 + 0.216i)16-s + (0.727 + 1.26i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.962495 + 0.477135i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.962495 + 0.477135i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (0.5 - 2.59i)T \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + (-5.5 - 9.52i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2 - 3.46i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 12T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + (-4 + 6.92i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.5 - 4.33i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.5 - 4.33i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (6 - 10.3i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (8 + 13.8i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + (-5 - 8.66i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.5 - 6.06i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 17T + 83T^{2} \) |
| 89 | \( 1 + (-6 + 10.3i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 13T + 97T^{2} \) |
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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
−10.79292762452724638664465111222, −9.941274010020743059097259597581, −8.867364660409077181261747633233, −8.440177060108751622383883953505, −7.22346425429358204649414479361, −6.42425224506268994397092301196, −5.53074109270997193839021520966, −4.71654687659452427119489782948, −2.83931838297286137220896364174, −1.29709934570008451130050918403,
0.871446442840109719682469124242, 2.78934340961851449924498546850, 3.76773780067947046869286477055, 4.84520803212155032273063532775, 6.05502022869840190829460552562, 7.22440246136445821184965402644, 7.969137781154364086106483468282, 9.399962436611779990352195139564, 9.820929896446834318321933931817, 10.59968536796956491800518294361