| L(s) = 1 | + (0.377 + 0.264i)2-s + (−0.611 − 1.67i)4-s + (−0.538 + 2.17i)5-s + (−1.52 − 3.26i)7-s + (0.451 − 1.68i)8-s + (−0.777 + 0.676i)10-s + (−2.85 − 3.40i)11-s + (0.226 + 0.323i)13-s + (0.288 − 1.63i)14-s + (−2.12 + 1.78i)16-s + (−1.03 − 3.86i)17-s + (1.05 − 0.610i)19-s + (3.97 − 0.421i)20-s + (−0.178 − 2.03i)22-s + (3.57 + 1.66i)23-s + ⋯ |
| L(s) = 1 | + (0.266 + 0.186i)2-s + (−0.305 − 0.839i)4-s + (−0.241 + 0.970i)5-s + (−0.575 − 1.23i)7-s + (0.159 − 0.596i)8-s + (−0.245 + 0.214i)10-s + (−0.860 − 1.02i)11-s + (0.0627 + 0.0896i)13-s + (0.0770 − 0.436i)14-s + (−0.530 + 0.445i)16-s + (−0.251 − 0.936i)17-s + (0.242 − 0.140i)19-s + (0.888 − 0.0942i)20-s + (−0.0380 − 0.434i)22-s + (0.746 + 0.347i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.251 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.251 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.585975 - 0.757425i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.585975 - 0.757425i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.538 - 2.17i)T \) |
| good | 2 | \( 1 + (-0.377 - 0.264i)T + (0.684 + 1.87i)T^{2} \) |
| 7 | \( 1 + (1.52 + 3.26i)T + (-4.49 + 5.36i)T^{2} \) |
| 11 | \( 1 + (2.85 + 3.40i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.226 - 0.323i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (1.03 + 3.86i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-1.05 + 0.610i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.57 - 1.66i)T + (14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (0.885 + 5.02i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-9.19 + 3.34i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (10.6 - 2.84i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-1.80 - 0.318i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (0.218 - 2.49i)T + (-42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (1.18 - 0.554i)T + (30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (-3.53 - 3.53i)T + 53iT^{2} \) |
| 59 | \( 1 + (-0.467 - 0.391i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (1.87 + 0.683i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-3.34 + 2.34i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (-10.9 - 6.30i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.00 - 1.07i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-11.8 + 2.09i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (1.54 - 2.20i)T + (-28.3 - 77.9i)T^{2} \) |
| 89 | \( 1 + (1.23 + 2.13i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.567 - 0.0496i)T + (95.5 + 16.8i)T^{2} \) |
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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
−10.82175044704897995988726746509, −10.22887972101213060854174270783, −9.450067899713204705018957459166, −8.026582565352648603309367531581, −7.01070941395287315343594026826, −6.36059700884820081047515103642, −5.19776502441649497411147221984, −3.98460304482037544208547070630, −2.88649881437348576015132234973, −0.56512280886366448659108174074,
2.21943262317393436463784996337, 3.45895406465383226832284665587, 4.74858765120266206437441683430, 5.41626994094993164834217000152, 6.89526927234373306420567084385, 8.118376735343112830803949560837, 8.681733218238475487067904060453, 9.490105915107510840661911872545, 10.67033964336532218973771095529, 12.01550434018311871671591044748
