L(s) = 1 | + 2-s + (1.5 − 0.866i)3-s + 4-s + (1.5 − 0.866i)6-s + 8-s + (1.5 − 2.59i)9-s + (1.5 + 2.59i)11-s + (1.5 − 0.866i)12-s + (−1 − 1.73i)13-s + 16-s + (1.5 − 2.59i)17-s + (1.5 − 2.59i)18-s + (0.5 + 0.866i)19-s + (1.5 + 2.59i)22-s + (3 − 5.19i)23-s + (1.5 − 0.866i)24-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (0.866 − 0.499i)3-s + 0.5·4-s + (0.612 − 0.353i)6-s + 0.353·8-s + (0.5 − 0.866i)9-s + (0.452 + 0.783i)11-s + (0.433 − 0.249i)12-s + (−0.277 − 0.480i)13-s + 0.250·16-s + (0.363 − 0.630i)17-s + (0.353 − 0.612i)18-s + (0.114 + 0.198i)19-s + (0.319 + 0.553i)22-s + (0.625 − 1.08i)23-s + (0.306 − 0.176i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 + 0.540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.841 + 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.17319 - 0.932049i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.17319 - 0.932049i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (-1.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + (6 - 10.3i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 3T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 - 5T + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + (5.5 - 9.52i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + (6 - 10.3i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.5 - 4.33i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.973740158568599808165891965718, −9.187593106459345513386212339656, −8.310605276390878168261327514320, −7.16143522311662487577991187545, −6.98234093902369285393060980017, −5.61326356056976060480773403085, −4.62893596945869284610312352694, −3.52814471671964207722507487007, −2.66618941877582088921449385903, −1.42899325781132999807970473280,
1.74165258904569940954104414321, 3.02419902868192752170812323228, 3.78890154961404174102545011959, 4.70909731124347955715070624828, 5.71398731471673298124068200625, 6.74871591616088355860649585528, 7.72569731220448094664001112760, 8.522296638279674604940960935764, 9.413379541671919443440134293759, 10.13392241563349590928045444277