L(s) = 1 | + (−0.866 − 0.5i)3-s + (0.866 − 2.5i)7-s + (0.499 + 0.866i)9-s + (2 − 3.46i)11-s − 7i·13-s + (5.19 + 3i)17-s + (1.5 + 2.59i)19-s + (−2 + 1.73i)21-s + (1.73 − i)23-s − 0.999i·27-s + 2·29-s + (−3.5 + 6.06i)31-s + (−3.46 + 1.99i)33-s + (−6.06 + 3.5i)37-s + (−3.5 + 6.06i)39-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.288i)3-s + (0.327 − 0.944i)7-s + (0.166 + 0.288i)9-s + (0.603 − 1.04i)11-s − 1.94i·13-s + (1.26 + 0.727i)17-s + (0.344 + 0.596i)19-s + (−0.436 + 0.377i)21-s + (0.361 − 0.208i)23-s − 0.192i·27-s + 0.371·29-s + (−0.628 + 1.08i)31-s + (−0.603 + 0.348i)33-s + (−0.996 + 0.575i)37-s + (−0.560 + 0.970i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.330 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.330 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.507179209\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.507179209\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.866 + 2.5i)T \) |
good | 11 | \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 7iT - 13T^{2} \) |
| 17 | \( 1 + (-5.19 - 3i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.5 - 2.59i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.73 + i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + (3.5 - 6.06i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (6.06 - 3.5i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 + 5iT - 43T^{2} \) |
| 47 | \( 1 + (-8.66 + 5i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6.92 + 4i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5 + 8.66i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3 - 5.19i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.59 + 1.5i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-12.9 - 7.5i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 8iT - 83T^{2} \) |
| 89 | \( 1 + (-1 - 1.73i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−8.534926252747354629595207590276, −8.146273525656206585298180564931, −7.32061433099054185289122197208, −6.52980699123046276067391203766, −5.54935496225336096582893441273, −5.16704232185747018864012424917, −3.67490527006674942546837791078, −3.27637809995456680316790594691, −1.46374658656284495258965082485, −0.63779283047354412179779196576,
1.42234121603893939128920430196, 2.40460427652870945629897663001, 3.72099769764717286966548204451, 4.65271578112438873857595804407, 5.22091761552953133000725145062, 6.19749145680771849829122434297, 6.96905034972331406931777393643, 7.62520503646788728978546054212, 8.885777034395081491964670095657, 9.357449481527709072157105107360