| L(s) = 1 | + (1.12 + 0.856i)2-s + (0.130 − 1.72i)3-s + (0.532 + 1.92i)4-s + (−0.588 − 0.214i)5-s + (1.62 − 1.83i)6-s + (4.57 + 0.806i)7-s + (−1.05 + 2.62i)8-s + (−2.96 − 0.450i)9-s + (−0.479 − 0.745i)10-s + (−1.37 − 3.76i)11-s + (3.39 − 0.669i)12-s + (0.583 + 0.695i)13-s + (4.45 + 4.82i)14-s + (−0.446 + 0.988i)15-s + (−3.43 + 2.05i)16-s + (2.52 − 1.45i)17-s + ⋯ |
| L(s) = 1 | + (0.795 + 0.605i)2-s + (0.0752 − 0.997i)3-s + (0.266 + 0.963i)4-s + (−0.263 − 0.0958i)5-s + (0.663 − 0.747i)6-s + (1.72 + 0.304i)7-s + (−0.371 + 0.928i)8-s + (−0.988 − 0.150i)9-s + (−0.151 − 0.235i)10-s + (−0.413 − 1.13i)11-s + (0.981 − 0.193i)12-s + (0.161 + 0.192i)13-s + (1.19 + 1.28i)14-s + (−0.115 + 0.255i)15-s + (−0.857 + 0.513i)16-s + (0.613 − 0.353i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.206i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 - 0.206i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.91180 + 0.199128i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.91180 + 0.199128i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.12 - 0.856i)T \) |
| 3 | \( 1 + (-0.130 + 1.72i)T \) |
| good | 5 | \( 1 + (0.588 + 0.214i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-4.57 - 0.806i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (1.37 + 3.76i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.583 - 0.695i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.52 + 1.45i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.50 - 4.34i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.819 - 4.64i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (6.21 + 5.21i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (4.29 - 0.756i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (3.72 - 2.15i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (5.41 + 6.44i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-5.48 + 1.99i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (1.38 - 7.85i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 3.06T + 53T^{2} \) |
| 59 | \( 1 + (0.537 - 1.47i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-3.32 - 0.586i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-9.02 + 7.57i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-3.19 - 5.53i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.18 - 3.78i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.52 + 6.58i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-6.42 + 7.65i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (1.88 + 1.08i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.312 - 0.113i)T + (74.3 - 62.3i)T^{2} \) |
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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
−12.35865135132403690482889722083, −11.63892203652884313853980857140, −11.02872074092578214942898608829, −8.765519089883740586045398068251, −8.009202759233576211367549550248, −7.51061491916353575755623861214, −5.94488036822966117960105836620, −5.30258396835558138170932559616, −3.70201182137478669073759592963, −2.00426433309781647984379746016,
2.07790419153617926054218480636, 3.77264929527216609964762196288, 4.74757067982967232722834473244, 5.38997739899115573857767914663, 7.21350994120852444465730940141, 8.442254506032925982774358793170, 9.679974332883028967777532539186, 10.77426061650872156378521018124, 11.08042626522420012532381137985, 12.13030350431002573848189160476
