| 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
−12.13030350431002573848189160476, −11.08042626522420012532381137985, −10.77426061650872156378521018124, −9.679974332883028967777532539186, −8.442254506032925982774358793170, −7.21350994120852444465730940141, −5.38997739899115573857767914663, −4.74757067982967232722834473244, −3.77264929527216609964762196288, −2.07790419153617926054218480636,
2.00426433309781647984379746016, 3.70201182137478669073759592963, 5.30258396835558138170932559616, 5.94488036822966117960105836620, 7.51061491916353575755623861214, 8.009202759233576211367549550248, 8.765519089883740586045398068251, 11.02872074092578214942898608829, 11.63892203652884313853980857140, 12.35865135132403690482889722083
