| L(s) = 1 | + (1.66 + 0.486i)3-s + (−2.11 + 1.77i)5-s + (−4.34 + 1.58i)7-s + (2.52 + 1.61i)9-s + (−2.54 − 2.13i)11-s + (0.625 + 3.54i)13-s + (−4.38 + 1.92i)15-s + (1.18 + 2.04i)17-s + (−1.11 + 1.93i)19-s + (−7.98 + 0.515i)21-s + (−2.65 − 0.965i)23-s + (0.459 − 2.60i)25-s + (3.41 + 3.91i)27-s + (1.17 − 6.63i)29-s + (8.26 + 3.00i)31-s + ⋯ |
| L(s) = 1 | + (0.959 + 0.280i)3-s + (−0.947 + 0.794i)5-s + (−1.64 + 0.597i)7-s + (0.842 + 0.538i)9-s + (−0.766 − 0.643i)11-s + (0.173 + 0.984i)13-s + (−1.13 + 0.496i)15-s + (0.286 + 0.496i)17-s + (−0.256 + 0.444i)19-s + (−1.74 + 0.112i)21-s + (−0.553 − 0.201i)23-s + (0.0919 − 0.521i)25-s + (0.657 + 0.753i)27-s + (0.217 − 1.23i)29-s + (1.48 + 0.540i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.456259 + 0.930331i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.456259 + 0.930331i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.66 - 0.486i)T \) |
| good | 5 | \( 1 + (2.11 - 1.77i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (4.34 - 1.58i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (2.54 + 2.13i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.625 - 3.54i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-1.18 - 2.04i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.11 - 1.93i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.65 + 0.965i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.17 + 6.63i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-8.26 - 3.00i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-3.78 - 6.55i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.644 - 3.65i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (4.43 + 3.72i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-7.67 + 2.79i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + 9.41T + 53T^{2} \) |
| 59 | \( 1 + (5.40 - 4.53i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-6.50 + 2.36i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-2.00 - 11.3i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (0.871 + 1.51i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (3.68 - 6.38i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.147 - 0.839i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (1.93 - 10.9i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (-6.35 + 11.0i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.46 - 1.23i)T + (16.8 + 95.5i)T^{2} \) |
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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
−11.48334571072172384785890619946, −10.28302559978209634390975615414, −9.804352335612257408444691851432, −8.635115628463989321504164167436, −7.990560676462941519800301479715, −6.85082225613268468480069754546, −6.05764574846407856083687546420, −4.24468829010363792759358379290, −3.33682087367434616920513725480, −2.60251939211535305510393002059,
0.57188597220582405635055718268, 2.78462221926512968649028183582, 3.67697424111119801880733080691, 4.73411874413738561383567883784, 6.32799042047917124876694505320, 7.43520740047512725367954402002, 7.934845702067832925924960131408, 9.016855934402655767588347227478, 9.811902866560391746333373889891, 10.57699588876549109180223273844
