| L(s) = 1 | + (0.324 − 1.70i)3-s + (2.22 − 0.392i)5-s + (1.16 − 2.02i)7-s + (−2.78 − 1.10i)9-s + (−2.52 + 1.45i)11-s + (−0.451 − 1.24i)13-s + (0.0536 − 3.91i)15-s + (−3.72 − 4.43i)17-s + (1.79 − 3.97i)19-s + (−3.06 − 2.64i)21-s + (8.06 + 1.42i)23-s + (0.112 − 0.0411i)25-s + (−2.78 + 4.38i)27-s + (1.64 + 1.38i)29-s + (−5.27 − 3.04i)31-s + ⋯ |
| L(s) = 1 | + (0.187 − 0.982i)3-s + (0.996 − 0.175i)5-s + (0.441 − 0.764i)7-s + (−0.929 − 0.367i)9-s + (−0.760 + 0.438i)11-s + (−0.125 − 0.344i)13-s + (0.0138 − 1.01i)15-s + (−0.902 − 1.07i)17-s + (0.412 − 0.910i)19-s + (−0.668 − 0.576i)21-s + (1.68 + 0.296i)23-s + (0.0225 − 0.00822i)25-s + (−0.535 + 0.844i)27-s + (0.305 + 0.256i)29-s + (−0.948 − 0.547i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.425 + 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.425 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.961556 - 1.51466i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.961556 - 1.51466i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.324 + 1.70i)T \) |
| 19 | \( 1 + (-1.79 + 3.97i)T \) |
| good | 5 | \( 1 + (-2.22 + 0.392i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-1.16 + 2.02i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.52 - 1.45i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.451 + 1.24i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (3.72 + 4.43i)T + (-2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (-8.06 - 1.42i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-1.64 - 1.38i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (5.27 + 3.04i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 2.98iT - 37T^{2} \) |
| 41 | \( 1 + (-8.52 - 3.10i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.0666 - 0.377i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (6.57 - 7.83i)T + (-8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (0.494 - 2.80i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-2.53 + 2.12i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.01 + 5.77i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-10.4 + 12.4i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (2.29 + 13.0i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-5.84 - 2.12i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (1.77 - 4.87i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-1.62 - 0.938i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (5.82 - 2.11i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-6.13 - 7.30i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−9.502821402334939351183849730772, −9.200843322671550694463606644827, −7.896956876376961163024045299483, −7.31156430787729574434931454586, −6.57491693411276829977328007381, −5.41335547178741619505996737592, −4.74978644775736415986832394350, −3.00186183666860718103184715035, −2.11274026867340088898930068113, −0.823758588351468139907316111916,
1.98048649842620435413944140509, 2.88693536283576113407474103891, 4.12677548328029016775932508354, 5.33135910635313320758518888888, 5.65920962872983142992599199831, 6.80905507195951517410499322624, 8.238586929626737585518204447773, 8.739266335851293720977428927799, 9.553295817326493916463948739157, 10.34867900944465086409777341722
