L(s) = 1 | + (−0.5 + 0.866i)2-s + (−1.73 − 0.0276i)3-s + (−0.499 − 0.866i)4-s + (−2.84 + 1.64i)5-s + (0.889 − 1.48i)6-s + (−0.683 + 2.55i)7-s + 0.999·8-s + (2.99 + 0.0958i)9-s − 3.28i·10-s − 2.07·11-s + (0.841 + 1.51i)12-s + (−3.53 + 0.716i)13-s + (−1.87 − 1.86i)14-s + (4.97 − 2.76i)15-s + (−0.5 + 0.866i)16-s + (−0.561 − 0.973i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.999 − 0.0159i)3-s + (−0.249 − 0.433i)4-s + (−1.27 + 0.735i)5-s + (0.363 − 0.606i)6-s + (−0.258 + 0.966i)7-s + 0.353·8-s + (0.999 + 0.0319i)9-s − 1.03i·10-s − 0.624·11-s + (0.243 + 0.436i)12-s + (−0.980 + 0.198i)13-s + (−0.500 − 0.499i)14-s + (1.28 − 0.714i)15-s + (−0.125 + 0.216i)16-s + (−0.136 − 0.236i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.531 + 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.531 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.121489 - 0.0671901i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.121489 - 0.0671901i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.73 + 0.0276i)T \) |
| 7 | \( 1 + (0.683 - 2.55i)T \) |
| 13 | \( 1 + (3.53 - 0.716i)T \) |
good | 5 | \( 1 + (2.84 - 1.64i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + 2.07T + 11T^{2} \) |
| 17 | \( 1 + (0.561 + 0.973i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + 1.01T + 19T^{2} \) |
| 23 | \( 1 + (-3.03 - 1.75i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-7.97 + 4.60i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.86 + 3.23i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (6.03 + 3.48i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-8.52 + 4.92i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.35 + 5.81i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.05 - 0.609i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (5.05 + 2.92i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (8.49 - 4.90i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 - 0.241iT - 61T^{2} \) |
| 67 | \( 1 - 11.8iT - 67T^{2} \) |
| 71 | \( 1 + (3.94 - 6.83i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (0.878 - 1.52i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.48 + 4.31i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 0.999iT - 83T^{2} \) |
| 89 | \( 1 + (6.59 + 3.80i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.21 + 5.57i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.67613894279831323003560300816, −9.886998267956431752507666924146, −8.794392306099681555355492993639, −7.66619083669438990468125698969, −7.13581391359997292772546086506, −6.18134874806701622713640063054, −5.20718063348285038703778320822, −4.22713927948730742318002014325, −2.62898947674518933013572426208, −0.12899311287221378759341188527,
1.03062801269681215108038026036, 3.19301916983618118409736565114, 4.53823416624179580854550242711, 4.81977897296634431003947219335, 6.57133015996113654545714558916, 7.51064133963114315783095651524, 8.155333092242515559827169259979, 9.370566171861357862854181021915, 10.41527171034018806918255055711, 10.83348597539482253344217948141