| L(s) = 1 | + (−0.366 + 1.36i)2-s + (0.874 + 2.86i)3-s + (−1.73 − i)4-s + (2.55 − 4.29i)5-s + (−4.24 + 0.144i)6-s + (6.45 + 2.70i)7-s + (2 − 1.99i)8-s + (−7.46 + 5.02i)9-s + (4.93 + 5.06i)10-s + (17.7 + 10.2i)11-s + (1.35 − 5.84i)12-s + (−10.2 + 10.2i)13-s + (−6.06 + 7.82i)14-s + (14.5 + 3.57i)15-s + (1.99 + 3.46i)16-s + (−4.39 − 16.4i)17-s + ⋯ |
| L(s) = 1 | + (−0.183 + 0.683i)2-s + (0.291 + 0.956i)3-s + (−0.433 − 0.250i)4-s + (0.511 − 0.859i)5-s + (−0.706 + 0.0241i)6-s + (0.922 + 0.386i)7-s + (0.250 − 0.249i)8-s + (−0.829 + 0.557i)9-s + (0.493 + 0.506i)10-s + (1.61 + 0.933i)11-s + (0.112 − 0.487i)12-s + (−0.790 + 0.790i)13-s + (−0.432 + 0.559i)14-s + (0.971 + 0.238i)15-s + (0.124 + 0.216i)16-s + (−0.258 − 0.965i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.241 - 0.970i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.241 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.06235 + 1.35983i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.06235 + 1.35983i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.366 - 1.36i)T \) |
| 3 | \( 1 + (-0.874 - 2.86i)T \) |
| 5 | \( 1 + (-2.55 + 4.29i)T \) |
| 7 | \( 1 + (-6.45 - 2.70i)T \) |
| good | 11 | \( 1 + (-17.7 - 10.2i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (10.2 - 10.2i)T - 169iT^{2} \) |
| 17 | \( 1 + (4.39 + 16.4i)T + (-250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (-11.5 - 20.0i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (20.5 + 5.50i)T + (458. + 264.5i)T^{2} \) |
| 29 | \( 1 + 2.30T + 841T^{2} \) |
| 31 | \( 1 + (-3.09 - 1.78i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-40.1 - 10.7i)T + (1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + 0.0268T + 1.68e3T^{2} \) |
| 43 | \( 1 + (6.00 + 6.00i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-11.8 - 3.18i)T + (1.91e3 + 1.10e3i)T^{2} \) |
| 53 | \( 1 + (-2.41 - 8.99i)T + (-2.43e3 + 1.40e3i)T^{2} \) |
| 59 | \( 1 + (13.7 + 7.94i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-41.9 + 24.2i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-6.37 - 23.7i)T + (-3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 + 80.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (36.9 + 137. i)T + (-4.61e3 + 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-22.7 + 13.1i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (83.7 + 83.7i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (39.3 - 22.7i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (84.5 + 84.5i)T + 9.40e3iT^{2} \) |
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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
−12.20708904511664863566172699735, −11.60221162567183854763905177725, −9.919766688544639910706598501520, −9.419590792511955688971756690741, −8.673226864955739497044135969922, −7.54339585972271907897923861783, −6.05262277999825298589214197270, −4.85968568869353212648320189838, −4.27287937001613360778170600522, −1.85782594716327749246642273446,
1.16222247084228453697034666787, 2.49700727100548754790128053899, 3.81701022699018763030062739294, 5.72455035649924781029769558866, 6.85993245540146231892159144814, 7.88087594131500533649953992359, 8.874777478115215744534086986845, 9.973637879716215336942592964938, 11.22057540294684857573045458373, 11.58814096538243013704020296795
