| L(s) = 1 | + (−1.49 + 2.25i)2-s + (−2.07 − 4.90i)4-s + (−3.86 + 1.23i)5-s + (−0.664 + 1.88i)7-s + (8.85 + 1.65i)8-s + (3.00 − 10.5i)10-s + (4.48 + 0.276i)11-s + (5.97 − 1.11i)13-s + (−3.26 − 4.31i)14-s + (−9.55 + 9.85i)16-s + (4.11 + 1.89i)17-s + (−4.30 − 2.31i)19-s + (14.0 + 16.4i)20-s + (−7.32 + 9.70i)22-s + (−1.49 + 2.99i)23-s + ⋯ |
| L(s) = 1 | + (−1.05 + 1.59i)2-s + (−1.03 − 2.45i)4-s + (−1.72 + 0.550i)5-s + (−0.251 + 0.712i)7-s + (3.12 + 0.585i)8-s + (0.950 − 3.34i)10-s + (1.35 + 0.0833i)11-s + (1.65 − 0.309i)13-s + (−0.871 − 1.15i)14-s + (−2.38 + 2.46i)16-s + (0.998 + 0.459i)17-s + (−0.987 − 0.530i)19-s + (3.14 + 3.67i)20-s + (−1.56 + 2.06i)22-s + (−0.311 + 0.624i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0693i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0693i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0197960 - 0.570560i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0197960 - 0.570560i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 103 | \( 1 + (-8.74 - 5.15i)T \) |
| good | 2 | \( 1 + (1.49 - 2.25i)T + (-0.779 - 1.84i)T^{2} \) |
| 5 | \( 1 + (3.86 - 1.23i)T + (4.08 - 2.88i)T^{2} \) |
| 7 | \( 1 + (0.664 - 1.88i)T + (-5.45 - 4.38i)T^{2} \) |
| 11 | \( 1 + (-4.48 - 0.276i)T + (10.9 + 1.35i)T^{2} \) |
| 13 | \( 1 + (-5.97 + 1.11i)T + (12.1 - 4.69i)T^{2} \) |
| 17 | \( 1 + (-4.11 - 1.89i)T + (11.0 + 12.9i)T^{2} \) |
| 19 | \( 1 + (4.30 + 2.31i)T + (10.4 + 15.8i)T^{2} \) |
| 23 | \( 1 + (1.49 - 2.99i)T + (-13.8 - 18.3i)T^{2} \) |
| 29 | \( 1 + (-0.0384 + 0.175i)T + (-26.3 - 12.1i)T^{2} \) |
| 31 | \( 1 + (-1.14 - 4.04i)T + (-26.3 + 16.3i)T^{2} \) |
| 37 | \( 1 + (-6.81 + 2.64i)T + (27.3 - 24.9i)T^{2} \) |
| 41 | \( 1 + (6.60 + 2.09i)T + (33.4 + 23.6i)T^{2} \) |
| 43 | \( 1 + (-0.447 - 2.88i)T + (-40.9 + 13.0i)T^{2} \) |
| 47 | \( 1 + (3.22 - 5.57i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.107 + 3.49i)T + (-52.8 + 3.26i)T^{2} \) |
| 59 | \( 1 + (2.00 + 5.68i)T + (-45.9 + 36.9i)T^{2} \) |
| 61 | \( 1 + (-1.25 - 13.5i)T + (-59.9 + 11.2i)T^{2} \) |
| 67 | \( 1 + (-1.54 + 1.79i)T + (-10.2 - 66.2i)T^{2} \) |
| 71 | \( 1 + (-0.592 - 2.70i)T + (-64.5 + 29.6i)T^{2} \) |
| 73 | \( 1 + (5.54 - 5.05i)T + (6.73 - 72.6i)T^{2} \) |
| 79 | \( 1 + (-3.21 - 2.93i)T + (7.28 + 78.6i)T^{2} \) |
| 83 | \( 1 + (6.44 + 7.52i)T + (-12.7 + 82.0i)T^{2} \) |
| 89 | \( 1 + (0.235 + 0.312i)T + (-24.3 + 85.6i)T^{2} \) |
| 97 | \( 1 + (4.12 - 1.89i)T + (63.1 - 73.6i)T^{2} \) |
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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
−10.31486217029446858234692164666, −9.208829872182421086119509294249, −8.533744648904333085253880724999, −8.089902576124512236716450189198, −7.17668403780151416747023678456, −6.43347519613760360570152624334, −5.83729054253652607648321400323, −4.40060739845351984267392908437, −3.52010616083701251619086430296, −1.11881035043066312346557474492,
0.54065687101436806872829004556, 1.42708330891242401801321641679, 3.33980741549627182066931197831, 3.89413416050640470038289821503, 4.37363364540354307465869558101, 6.55807272799766576017670659579, 7.62333510977137440620247056227, 8.351130946333141980776863768717, 8.761807327413550979544751624169, 9.720876214582756177087641932945
