| L(s) = 1 | + (0.0871 − 0.996i)2-s + (−0.984 − 0.173i)4-s + (0.0741 + 0.0345i)5-s + (−1.92 + 0.700i)7-s + (−0.258 + 0.965i)8-s + (0.0408 − 0.0708i)10-s + (−0.0467 − 0.0809i)11-s + (−3.00 − 4.29i)13-s + (0.530 + 1.97i)14-s + (0.939 + 0.342i)16-s + (2.94 − 4.21i)17-s + (−7.61 + 0.666i)19-s + (−0.0669 − 0.0468i)20-s + (−0.0847 + 0.0395i)22-s + (−6.72 + 1.80i)23-s + ⋯ |
| L(s) = 1 | + (0.0616 − 0.704i)2-s + (−0.492 − 0.0868i)4-s + (0.0331 + 0.0154i)5-s + (−0.727 + 0.264i)7-s + (−0.0915 + 0.341i)8-s + (0.0129 − 0.0223i)10-s + (−0.0140 − 0.0244i)11-s + (−0.833 − 1.19i)13-s + (0.141 + 0.529i)14-s + (0.234 + 0.0855i)16-s + (0.715 − 1.02i)17-s + (−1.74 + 0.152i)19-s + (−0.0149 − 0.0104i)20-s + (−0.0180 + 0.00842i)22-s + (−1.40 + 0.375i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.187i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.982 - 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0443399 + 0.468175i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0443399 + 0.468175i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0871 + 0.996i)T \) |
| 3 | \( 1 \) |
| 37 | \( 1 + (6.08 + 0.146i)T \) |
| good | 5 | \( 1 + (-0.0741 - 0.0345i)T + (3.21 + 3.83i)T^{2} \) |
| 7 | \( 1 + (1.92 - 0.700i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (0.0467 + 0.0809i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (3.00 + 4.29i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (-2.94 + 4.21i)T + (-5.81 - 15.9i)T^{2} \) |
| 19 | \( 1 + (7.61 - 0.666i)T + (18.7 - 3.29i)T^{2} \) |
| 23 | \( 1 + (6.72 - 1.80i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-3.03 - 0.813i)T + (25.1 + 14.5i)T^{2} \) |
| 31 | \( 1 + (0.356 + 0.356i)T + 31iT^{2} \) |
| 41 | \( 1 + (0.847 - 4.80i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-5.17 + 5.17i)T - 43iT^{2} \) |
| 47 | \( 1 + (-7.78 - 4.49i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.03 + 5.57i)T + (-40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (3.74 + 8.02i)T + (-37.9 + 45.1i)T^{2} \) |
| 61 | \( 1 + (4.71 - 3.30i)T + (20.8 - 57.3i)T^{2} \) |
| 67 | \( 1 + (-1.89 - 5.19i)T + (-51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (6.74 - 8.03i)T + (-12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + 4.72iT - 73T^{2} \) |
| 79 | \( 1 + (-5.96 + 12.7i)T + (-50.7 - 60.5i)T^{2} \) |
| 83 | \( 1 + (-11.2 + 1.97i)T + (77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (5.09 - 2.37i)T + (57.2 - 68.1i)T^{2} \) |
| 97 | \( 1 + (-0.309 - 1.15i)T + (-84.0 + 48.5i)T^{2} \) |
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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
−10.12892299800449118351099529290, −9.480164935579498637010334511778, −8.405296672283928636469841890573, −7.58481135663173512934574084062, −6.30714915368743356450811219461, −5.46167355843715384840458740520, −4.31549532993641149289436657835, −3.16488043863363606928252086946, −2.22707358254215702149217997133, −0.22964306331364436031938028571,
2.08351206386321628159675042440, 3.75329692724989981046601765795, 4.48004757456554392187967698463, 5.83867581768252846693972360491, 6.50932158918030155536158323569, 7.35671057682421965028574225570, 8.321395001504077670681811988523, 9.177956126990232658955507237817, 10.02365645956319350730931726252, 10.72209254345196358494252260814
