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 + (1.5 + 2.59i)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 + (1 + 1.73i)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 + (0.452 + 0.783i)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 + (0.242 + 0.420i)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
\(L(1)\) |
\(\approx\) |
\(1.44192 + 0.714803i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.44192 + 0.714803i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4 + 6.92i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.5 - 6.06i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + (3 + 5.19i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.5 + 11.2i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 3T + 83T^{2} \) |
| 89 | \( 1 + (-4 + 6.92i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 15T + 97T^{2} \) |
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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
−10.99263051115799243784621245902, −9.969715769515208000355723223381, −9.440659719420773425379446678364, −8.503318192651798780981836354118, −7.36311552992085832989946812460, −6.14698395014221225722082117442, −5.17728991315345883317191745758, −4.09037409148173204379043917726, −3.09418854216884818716346871031, −1.98424556898645992557921746343,
0.838685091202834542566605555383, 2.90760476207824051830328898095, 4.04653632235164986751997838867, 4.99480724927504952447009825478, 6.40653738964839307384587513408, 6.86882025835806760698301686441, 8.080000712262077456968422914782, 8.486052658381911535361474971829, 9.690002701981892593660266498012, 10.65168223376858963665293482734