L(s) = 1 | + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)11-s − 1.73i·17-s − 1.73i·19-s + (0.5 + 0.866i)25-s + 0.999·27-s + (−0.499 + 0.866i)33-s + (−1.5 − 0.866i)41-s + (−1.5 + 0.866i)43-s + (0.5 − 0.866i)49-s + (−1.49 + 0.866i)51-s + (−1.49 + 0.866i)57-s + (0.5 − 0.866i)59-s + (1.5 + 0.866i)67-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)11-s − 1.73i·17-s − 1.73i·19-s + (0.5 + 0.866i)25-s + 0.999·27-s + (−0.499 + 0.866i)33-s + (−1.5 − 0.866i)41-s + (−1.5 + 0.866i)43-s + (0.5 − 0.866i)49-s + (−1.49 + 0.866i)51-s + (−1.49 + 0.866i)57-s + (0.5 − 0.866i)59-s + (1.5 + 0.866i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7444951891\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7444951891\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
good | 5 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + 1.73iT - T^{2} \) |
| 19 | \( 1 + 1.73iT - T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−9.713698889122175401365338352956, −8.807684935307733772315429504973, −8.059808779249993604386062059198, −7.00871047801996196995493506003, −6.70528334974994246019662780858, −5.26936110194668502827172135590, −5.06196629728256267700134865641, −3.26355754649610131439298947658, −2.35311361942562411350292863515, −0.71063606592494478887874750811,
1.82502763960839879152377636775, 3.38229133065129211881688170383, 4.19236168335300638227054785059, 5.09349761546224666032753480224, 5.97533084442321874325234846217, 6.70989330109287694797106737161, 8.005446621875925014411523275044, 8.561260906443996324119220963297, 9.729923335212448247062228102154, 10.30401613582429353912948649270