L(s) = 1 | + (0.951 − 0.309i)2-s + (0.809 − 0.587i)4-s + (−2.23 + 0.0466i)5-s − 3.52i·7-s + (0.587 − 0.809i)8-s + (−2.11 + 0.735i)10-s + (−1.62 − 4.99i)11-s + (−0.588 − 0.191i)13-s + (−1.08 − 3.34i)14-s + (0.309 − 0.951i)16-s + (−2.02 + 2.78i)17-s + (1.83 + 1.33i)19-s + (−1.78 + 1.35i)20-s + (−3.08 − 4.25i)22-s + (8.51 − 2.76i)23-s + ⋯ |
L(s) = 1 | + (0.672 − 0.218i)2-s + (0.404 − 0.293i)4-s + (−0.999 + 0.0208i)5-s − 1.33i·7-s + (0.207 − 0.286i)8-s + (−0.667 + 0.232i)10-s + (−0.489 − 1.50i)11-s + (−0.163 − 0.0530i)13-s + (−0.290 − 0.895i)14-s + (0.0772 − 0.237i)16-s + (−0.491 + 0.676i)17-s + (0.422 + 0.306i)19-s + (−0.398 + 0.302i)20-s + (−0.658 − 0.906i)22-s + (1.77 − 0.576i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.208 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.208 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.969456 - 1.19742i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.969456 - 1.19742i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.951 + 0.309i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.23 - 0.0466i)T \) |
good | 7 | \( 1 + 3.52iT - 7T^{2} \) |
| 11 | \( 1 + (1.62 + 4.99i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (0.588 + 0.191i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (2.02 - 2.78i)T + (-5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.83 - 1.33i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-8.51 + 2.76i)T + (18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-2.16 + 1.57i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (7.90 + 5.74i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (0.952 + 0.309i)T + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (1.94 - 5.98i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 1.51iT - 43T^{2} \) |
| 47 | \( 1 + (-6.27 - 8.63i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (0.325 + 0.447i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.0861 + 0.265i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-1.13 - 3.50i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (7.00 - 9.63i)T + (-20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (-3.84 + 2.79i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-9.35 + 3.03i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-5.27 + 3.83i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-1.56 + 2.15i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (4.13 + 12.7i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (2.56 + 3.53i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−10.97398613552715281277999519446, −10.42016630006285267204630813135, −8.940199072394315081561906577161, −7.897334369572400576352133207334, −7.17479542574435868002265670771, −6.08140783521985676710330766784, −4.80278677543492500750383298004, −3.86600225106217591947179762590, −3.03743513789464361714261164363, −0.793516707318753513436134825722,
2.28741300768822514503708549684, 3.41192713819974300584681897356, 4.88457312315378130909676796926, 5.25142881281840208783080894552, 6.96164729730794461630356024606, 7.33055901125390993446425841880, 8.642627534718507318721378053933, 9.335225266319681354046349375106, 10.73442196095117412147845526185, 11.57435682005150765934834226716