| L(s) = 1 | + (−2.44 − 0.213i)2-s + (3.96 + 0.698i)4-s + (−1.02 − 1.98i)5-s + (−3.87 − 2.71i)7-s + (−4.80 − 1.28i)8-s + (2.07 + 5.08i)10-s + (−0.927 + 2.54i)11-s + (0.0885 + 1.01i)13-s + (8.89 + 7.46i)14-s + (3.89 + 1.41i)16-s + (1.15 − 0.309i)17-s + (0.507 − 0.292i)19-s + (−2.66 − 8.59i)20-s + (2.81 − 6.03i)22-s + (0.750 + 1.07i)23-s + ⋯ |
| L(s) = 1 | + (−1.72 − 0.151i)2-s + (1.98 + 0.349i)4-s + (−0.457 − 0.889i)5-s + (−1.46 − 1.02i)7-s + (−1.69 − 0.454i)8-s + (0.655 + 1.60i)10-s + (−0.279 + 0.768i)11-s + (0.0245 + 0.280i)13-s + (2.37 + 1.99i)14-s + (0.974 + 0.354i)16-s + (0.279 − 0.0749i)17-s + (0.116 − 0.0671i)19-s + (−0.595 − 1.92i)20-s + (0.599 − 1.28i)22-s + (0.156 + 0.223i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.184 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.184 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0738023 + 0.0889112i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0738023 + 0.0889112i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (1.02 + 1.98i)T \) |
| good | 2 | \( 1 + (2.44 + 0.213i)T + (1.96 + 0.347i)T^{2} \) |
| 7 | \( 1 + (3.87 + 2.71i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (0.927 - 2.54i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.0885 - 1.01i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (-1.15 + 0.309i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-0.507 + 0.292i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.750 - 1.07i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (-0.185 + 0.155i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.978 - 5.54i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (0.227 + 0.850i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-2.80 + 3.33i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-4.67 - 10.0i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (3.77 - 5.39i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (6.73 - 6.73i)T - 53iT^{2} \) |
| 59 | \( 1 + (11.1 - 4.07i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (1.40 + 7.98i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (7.77 - 0.680i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (1.13 + 0.654i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.567 - 2.11i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-2.17 - 2.59i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.839 + 9.59i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (-1.54 - 2.66i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (3.04 - 1.41i)T + (62.3 - 74.3i)T^{2} \) |
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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
−11.13672266883736210841539721323, −10.32758271872270503909892954376, −9.516390555091509424030734805160, −9.092251580338315917782029924645, −7.80033810004065735730815548629, −7.30499652620292730921040327727, −6.28868862067222294906783633936, −4.50792030768521996162788449925, −3.11051670037890664185512190650, −1.27277667549810216174036606609,
0.14002109513411742260699057692, 2.47330198068974705153190824116, 3.37654308007093322849500248151, 5.85106522114571273182869920221, 6.49831614232568843748241627820, 7.46992496677365030002564176749, 8.356430032126318470210704497314, 9.204764807321603875896584523545, 9.961143710082922750302376851543, 10.69905621626604267712330318572
