L(s) = 1 | + (0.5 − 0.866i)2-s − 3-s + (−0.499 − 0.866i)4-s + (−2.07 − 3.58i)5-s + (−0.5 + 0.866i)6-s + (2.11 − 1.59i)7-s − 0.999·8-s + 9-s − 4.14·10-s + 0.523·11-s + (0.499 + 0.866i)12-s + (−3.28 + 1.47i)13-s + (−0.321 − 2.62i)14-s + (2.07 + 3.58i)15-s + (−0.5 + 0.866i)16-s + (1.26 + 2.18i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s − 0.577·3-s + (−0.249 − 0.433i)4-s + (−0.926 − 1.60i)5-s + (−0.204 + 0.353i)6-s + (0.798 − 0.601i)7-s − 0.353·8-s + 0.333·9-s − 1.30·10-s + 0.157·11-s + (0.144 + 0.249i)12-s + (−0.912 + 0.409i)13-s + (−0.0859 − 0.701i)14-s + (0.534 + 0.926i)15-s + (−0.125 + 0.216i)16-s + (0.305 + 0.529i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 - 0.294i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.955 - 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.123591 + 0.820941i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.123591 + 0.820941i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + (-2.11 + 1.59i)T \) |
| 13 | \( 1 + (3.28 - 1.47i)T \) |
good | 5 | \( 1 + (2.07 + 3.58i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 - 0.523T + 11T^{2} \) |
| 17 | \( 1 + (-1.26 - 2.18i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + 5.69T + 19T^{2} \) |
| 23 | \( 1 + (-3.69 + 6.40i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.54 + 2.66i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2.17 - 3.76i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.83 + 4.90i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.33 - 4.03i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.81 - 8.34i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (5.58 + 9.66i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.00192 - 0.00332i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.05 + 7.02i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 - 6.01T + 61T^{2} \) |
| 67 | \( 1 - 3.23T + 67T^{2} \) |
| 71 | \( 1 + (-3.98 + 6.90i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.99 + 10.3i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.15 - 2.00i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 3.08T + 83T^{2} \) |
| 89 | \( 1 + (-5.42 + 9.39i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.31 - 2.28i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.62009208270330813645751147289, −9.498329386801067467175906946150, −8.533711721435758681792111997520, −7.83158189682054647031186661700, −6.57443657646155099373415841689, −5.09630223569029501515954070673, −4.63979861829907439925260688108, −3.89889002283052094607472285230, −1.77558295864066770436406442467, −0.46159957991740011945290513467,
2.52623540578326875931186483580, 3.70643146179463328657720337125, 4.85141382188606782085987890306, 5.81861233124943009393143467706, 6.88361900167409781853018484493, 7.45234084814992842012951241898, 8.246225080043922870744325098291, 9.580129425840139947613126915602, 10.70109465135718458130933476341, 11.34803054560201382947794046053