| 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
−10.34867900944465086409777341722, −9.553295817326493916463948739157, −8.739266335851293720977428927799, −8.238586929626737585518204447773, −6.80905507195951517410499322624, −5.65920962872983142992599199831, −5.33135910635313320758518888888, −4.12677548328029016775932508354, −2.88693536283576113407474103891, −1.98048649842620435413944140509,
0.823758588351468139907316111916, 2.11274026867340088898930068113, 3.00186183666860718103184715035, 4.74978644775736415986832394350, 5.41335547178741619505996737592, 6.57491693411276829977328007381, 7.31156430787729574434931454586, 7.896956876376961163024045299483, 9.200843322671550694463606644827, 9.502821402334939351183849730772
