| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.504 + 0.863i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.504 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0607346 - 0.105816i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0607346 - 0.105816i\) |
| \(L(\frac{3}{2})\) |
|
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
−10.30454139234502379735029409580, −9.138161252196721263449431734717, −8.515886089941952828331004826780, −7.56566328191415845821758683239, −6.58772540005518291831346378577, −5.90530258560258323455838659360, −4.82624641696300917259844412181, −3.72354523203762854719788245739, −2.35430810442712448524708388259, −0.06205091880159728981061419921,
1.92058443379193883394150000581, 3.07627691810482546395225523195, 4.23074326290558734506339168082, 5.07687493381796572457420959505, 6.57331725480300478108839239123, 7.13860620604290832959387633241, 8.450611852686809585880247563169, 9.446252624591708368820744818040, 9.695948283347492818809899145446, 11.17635245592663955431343007572
