L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.0338 − 0.0585i)5-s + (1.44 − 2.21i)7-s − 0.999i·8-s + (−0.0585 − 0.0338i)10-s + (3.40 + 1.96i)11-s + 3.84i·13-s + (0.142 − 2.64i)14-s + (−0.5 − 0.866i)16-s + (0.775 − 1.34i)17-s + (5.06 − 2.92i)19-s − 0.0676·20-s + 3.92·22-s + (−4.78 + 2.76i)23-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.0151 − 0.0261i)5-s + (0.545 − 0.837i)7-s − 0.353i·8-s + (−0.0185 − 0.0106i)10-s + (1.02 + 0.592i)11-s + 1.06i·13-s + (0.0380 − 0.706i)14-s + (−0.125 − 0.216i)16-s + (0.188 − 0.325i)17-s + (1.16 − 0.670i)19-s − 0.0151·20-s + 0.837·22-s + (−0.998 + 0.576i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.624145688\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.624145688\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.44 + 2.21i)T \) |
good | 5 | \( 1 + (0.0338 + 0.0585i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.40 - 1.96i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 3.84iT - 13T^{2} \) |
| 17 | \( 1 + (-0.775 + 1.34i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.06 + 2.92i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.78 - 2.76i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 1.39iT - 29T^{2} \) |
| 31 | \( 1 + (1.09 + 0.632i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.35 + 7.54i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 + 1.47T + 43T^{2} \) |
| 47 | \( 1 + (-1.77 - 3.06i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.28 + 3.63i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.70 - 8.14i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.0705 - 0.0407i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.67 + 13.2i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 4.30iT - 71T^{2} \) |
| 73 | \( 1 + (-6.12 - 3.53i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.42 - 5.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 7.86T + 83T^{2} \) |
| 89 | \( 1 + (-5.84 - 10.1i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 0.419iT - 97T^{2} \) |
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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
−9.625524425203271177658217232322, −9.205561164706063357211941693868, −7.83385016705637477639053919940, −7.11203420078697938218635995432, −6.36676406819882137436028717292, −5.18140631314782990432409288997, −4.32102037219501967748677168415, −3.72862924992959177817902442826, −2.24826075406485869985171907409, −1.13273051938404044251932205354,
1.45831342873739133079650463774, 2.93287635579867041347313797023, 3.74125332579165863515900694163, 4.96266719780898986324337548328, 5.70472074393770401782755039396, 6.33962633779128424290524951700, 7.51757405212152721149751817990, 8.214421793830219558713236646388, 8.938942381377202262282492403120, 9.910114626157780963715319633210