| L(s) = 1 | + (1.43 + 0.972i)3-s + (−3.94 − 1.43i)5-s + (−0.610 + 3.46i)7-s + (1.10 + 2.78i)9-s + (−1.73 + 0.632i)11-s + (−1.78 + 1.49i)13-s + (−4.26 − 5.89i)15-s + (−0.799 − 1.38i)17-s + (−2.31 + 4.00i)19-s + (−4.24 + 4.36i)21-s + (0.308 + 1.74i)23-s + (9.68 + 8.12i)25-s + (−1.12 + 5.07i)27-s + (0.882 + 0.740i)29-s + (−0.322 − 1.83i)31-s + ⋯ |
| L(s) = 1 | + (0.827 + 0.561i)3-s + (−1.76 − 0.642i)5-s + (−0.230 + 1.30i)7-s + (0.369 + 0.929i)9-s + (−0.523 + 0.190i)11-s + (−0.495 + 0.415i)13-s + (−1.10 − 1.52i)15-s + (−0.194 − 0.336i)17-s + (−0.530 + 0.918i)19-s + (−0.925 + 0.953i)21-s + (0.0642 + 0.364i)23-s + (1.93 + 1.62i)25-s + (−0.215 + 0.976i)27-s + (0.163 + 0.137i)29-s + (−0.0579 − 0.328i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.697 - 0.716i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.697 - 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.331416 + 0.784705i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.331416 + 0.784705i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.43 - 0.972i)T \) |
| good | 5 | \( 1 + (3.94 + 1.43i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (0.610 - 3.46i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (1.73 - 0.632i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (1.78 - 1.49i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.799 + 1.38i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.31 - 4.00i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.308 - 1.74i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.882 - 0.740i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (0.322 + 1.83i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (4.38 + 7.59i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.98 + 2.50i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.41 + 0.880i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (1.29 - 7.35i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 8.02T + 53T^{2} \) |
| 59 | \( 1 + (-1.15 - 0.418i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (0.754 - 4.28i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-4.86 + 4.08i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (0.871 + 1.50i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (1.37 - 2.38i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.63 - 6.40i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (8.65 + 7.25i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (2.71 - 4.69i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-11.3 + 4.13i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.60669600843594643498094665372, −10.60882583893905929496196298005, −9.352013818042793189245824602133, −8.775701515373259168479354875484, −8.000372909957883748053140417040, −7.26323968744306459030692389242, −5.48269453710103445289964911066, −4.51086658221235314216722470368, −3.61581817949731594249005974016, −2.37953475338763774853710899864,
0.47814752169453852311833610508, 2.79501008934557966627224325979, 3.67889997033067701735128070924, 4.54796510800980946442907656515, 6.71075335899894517870112158299, 7.17760884830961734824831708688, 7.966043632483752742586329289602, 8.609703746791648187669461473385, 10.11816678275971116150299705603, 10.79557877469356056446520805710
