L(s) = 1 | + (2.13 − 0.571i)2-s + (1.67 + 0.448i)3-s + (0.756 − 0.436i)4-s + (2.78 + 4.15i)5-s + 3.82·6-s + (2.73 − 6.44i)7-s + (−4.88 + 4.88i)8-s + (2.59 + 1.50i)9-s + (8.31 + 7.26i)10-s + (−4.69 − 8.12i)11-s + (1.46 − 0.391i)12-s + (−0.405 + 0.405i)13-s + (2.15 − 15.3i)14-s + (2.79 + 8.19i)15-s + (−9.36 + 16.2i)16-s + (2.82 − 10.5i)17-s + ⋯ |
L(s) = 1 | + (1.06 − 0.285i)2-s + (0.557 + 0.149i)3-s + (0.189 − 0.109i)4-s + (0.557 + 0.830i)5-s + 0.637·6-s + (0.390 − 0.920i)7-s + (−0.610 + 0.610i)8-s + (0.288 + 0.166i)9-s + (0.831 + 0.726i)10-s + (−0.426 − 0.738i)11-s + (0.121 − 0.0326i)12-s + (−0.0311 + 0.0311i)13-s + (0.153 − 1.09i)14-s + (0.186 + 0.546i)15-s + (−0.585 + 1.01i)16-s + (0.166 − 0.621i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0208i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.53853 + 0.0264045i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.53853 + 0.0264045i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.67 - 0.448i)T \) |
| 5 | \( 1 + (-2.78 - 4.15i)T \) |
| 7 | \( 1 + (-2.73 + 6.44i)T \) |
good | 2 | \( 1 + (-2.13 + 0.571i)T + (3.46 - 2i)T^{2} \) |
| 11 | \( 1 + (4.69 + 8.12i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (0.405 - 0.405i)T - 169iT^{2} \) |
| 17 | \( 1 + (-2.82 + 10.5i)T + (-250. - 144.5i)T^{2} \) |
| 19 | \( 1 + (23.8 + 13.7i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-1.02 - 3.81i)T + (-458. + 264.5i)T^{2} \) |
| 29 | \( 1 - 34.6iT - 841T^{2} \) |
| 31 | \( 1 + (11.8 + 20.4i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-25.6 + 6.86i)T + (1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 - 54.6T + 1.68e3T^{2} \) |
| 43 | \( 1 + (47.3 - 47.3i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (16.9 - 4.55i)T + (1.91e3 - 1.10e3i)T^{2} \) |
| 53 | \( 1 + (-14.7 - 3.95i)T + (2.43e3 + 1.40e3i)T^{2} \) |
| 59 | \( 1 + (-90.3 + 52.1i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (12.5 - 21.7i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-0.364 + 1.36i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 - 86.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + (84.9 + 22.7i)T + (4.61e3 + 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-128. - 74.1i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-35.5 + 35.5i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (-130. - 75.1i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-74.8 - 74.8i)T + 9.40e3iT^{2} \) |
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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
−13.50246991452089202055315867169, −13.01805072148110587464693684759, −11.32677890425682963801255767493, −10.67149255781752636329695409323, −9.290618626535639528014095854400, −7.947309328770989942758243035942, −6.56129284206556171233112386540, −5.09340424575573959151660763335, −3.76436366736412027314598896122, −2.56529055245410112651385257442,
2.19096069793402111425857903784, 4.17887507140922445692307541654, 5.29505859368181476822462418780, 6.30902853456022444505732977489, 8.100410605313597915033677441780, 9.073977444701743223464930451672, 10.13837553257817061352344557104, 12.08437932718170273558435193368, 12.72584334838124959036019772543, 13.43127739647490963329124145235