L(s) = 1 | + (−0.809 + 0.587i)2-s + (0.309 + 0.951i)3-s + (0.309 − 0.951i)4-s + (3.23 + 2.35i)5-s + (−0.809 − 0.587i)6-s + (−0.618 + 1.90i)7-s + (0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s − 4·10-s + 12-s + (−3.23 + 2.35i)13-s + (−0.618 − 1.90i)14-s + (−1.23 + 3.80i)15-s + (−0.809 − 0.587i)16-s + (1.61 + 1.17i)17-s + (0.309 − 0.951i)18-s + ⋯ |
L(s) = 1 | + (−0.572 + 0.415i)2-s + (0.178 + 0.549i)3-s + (0.154 − 0.475i)4-s + (1.44 + 1.05i)5-s + (−0.330 − 0.239i)6-s + (−0.233 + 0.718i)7-s + (0.109 + 0.336i)8-s + (−0.269 + 0.195i)9-s − 1.26·10-s + 0.288·12-s + (−0.897 + 0.652i)13-s + (−0.165 − 0.508i)14-s + (−0.319 + 0.982i)15-s + (−0.202 − 0.146i)16-s + (0.392 + 0.285i)17-s + (0.0728 − 0.224i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.751 - 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.751 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.481385 + 1.27922i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.481385 + 1.27922i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.809 - 0.587i)T \) |
| 3 | \( 1 + (-0.309 - 0.951i)T \) |
| 11 | \( 1 \) |
good | 5 | \( 1 + (-3.23 - 2.35i)T + (1.54 + 4.75i)T^{2} \) |
| 7 | \( 1 + (0.618 - 1.90i)T + (-5.66 - 4.11i)T^{2} \) |
| 13 | \( 1 + (3.23 - 2.35i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.61 - 1.17i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + (-3.09 + 9.51i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-6.47 + 4.70i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (0.618 - 1.90i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-0.618 - 1.90i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (0.618 + 1.90i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (3.23 - 2.35i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-6.47 - 4.70i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 + (1.61 + 1.17i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (1.85 - 5.70i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (8.09 - 5.87i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (3.23 + 2.35i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + (-1.61 + 1.17i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.10400732649269491074459976356, −9.992390160930914284420106764602, −9.290217279351539339308340527595, −8.254318846369647746861897167618, −7.20100803779847773133243923953, −6.07639380052804603894934918786, −5.87660365934706755878391445563, −4.45586196387025145687576414119, −2.76733570904848788870478417095, −2.09849354589634995346058730785,
0.829690035986764433367018455699, 1.89891918537028913759514080020, 3.05317059375928460858040905301, 4.65260034158783467260069562989, 5.61861061913938010939195723787, 6.64779194400396546638654646058, 7.60378266507085922403310895118, 8.520730771068687782028426463321, 9.263110855819822031129834274695, 10.11594482489389953588824125819