L(s) = 1 | + (0.5 + 0.866i)2-s − 3-s + (−0.499 + 0.866i)4-s + (0.114 − 0.197i)5-s + (−0.5 − 0.866i)6-s + (0.848 − 2.50i)7-s − 0.999·8-s + 9-s + 0.228·10-s + 3.41·11-s + (0.499 − 0.866i)12-s + (−1.62 + 3.21i)13-s + (2.59 − 0.518i)14-s + (−0.114 + 0.197i)15-s + (−0.5 − 0.866i)16-s + (2.70 − 4.68i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s − 0.577·3-s + (−0.249 + 0.433i)4-s + (0.0509 − 0.0883i)5-s + (−0.204 − 0.353i)6-s + (0.320 − 0.947i)7-s − 0.353·8-s + 0.333·9-s + 0.0721·10-s + 1.02·11-s + (0.144 − 0.249i)12-s + (−0.451 + 0.892i)13-s + (0.693 − 0.138i)14-s + (−0.0294 + 0.0509i)15-s + (−0.125 − 0.216i)16-s + (0.656 − 1.13i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.49960 + 0.439893i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.49960 + 0.439893i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.848 + 2.50i)T \) |
| 13 | \( 1 + (1.62 - 3.21i)T \) |
good | 5 | \( 1 + (-0.114 + 0.197i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 - 3.41T + 11T^{2} \) |
| 17 | \( 1 + (-2.70 + 4.68i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 - 6.34T + 19T^{2} \) |
| 23 | \( 1 + (-0.959 - 1.66i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.851 + 1.47i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.78 + 3.08i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.09 - 3.62i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.59 + 2.75i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.17 - 8.97i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (5.57 - 9.65i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.31 + 5.74i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-7.38 + 12.7i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + 6.71T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 + (0.0390 + 0.0677i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (6.31 + 10.9i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.811 + 1.40i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 1.70T + 83T^{2} \) |
| 89 | \( 1 + (8.77 + 15.1i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.82 - 11.8i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.30327054108929564998792543365, −9.675055506833174866399249159112, −9.397786730907442253612178136107, −7.80283760753350300902413073019, −7.20346265830511067918135240170, −6.40922867739194805293005125874, −5.20472399541815381706031323076, −4.49251897488558736122646384453, −3.35456553261346161861713284635, −1.18423878233764156723338362058,
1.26052798868033874402035704239, 2.75146418175952106417266364340, 3.96594555080335639466954088774, 5.24609107263644310091375227487, 5.78090241961830918285615333942, 6.91686206474718405307754818764, 8.181261282445747477591025419102, 9.132821911319771987559664481773, 10.05712847543126562627924751418, 10.79480001168168946803668535388