| L(s) = 1 | + (−0.0539 − 0.00813i)2-s + (0.826 − 0.563i)3-s + (−1.90 − 0.588i)4-s + (−0.234 − 3.12i)5-s + (−0.0491 + 0.0236i)6-s + (−2.39 − 1.13i)7-s + (0.196 + 0.0946i)8-s + (0.365 − 0.930i)9-s + (−0.0127 + 0.170i)10-s + (1.33 + 3.39i)11-s + (−1.90 + 0.588i)12-s + (2.60 − 3.26i)13-s + (0.119 + 0.0805i)14-s + (−1.95 − 2.44i)15-s + (3.29 + 2.24i)16-s + (5.16 + 4.79i)17-s + ⋯ |
| L(s) = 1 | + (−0.0381 − 0.00575i)2-s + (0.477 − 0.325i)3-s + (−0.954 − 0.294i)4-s + (−0.104 − 1.39i)5-s + (−0.0200 + 0.00966i)6-s + (−0.903 − 0.427i)7-s + (0.0694 + 0.0334i)8-s + (0.121 − 0.310i)9-s + (−0.00404 + 0.0539i)10-s + (0.401 + 1.02i)11-s + (−0.550 + 0.169i)12-s + (0.722 − 0.906i)13-s + (0.0320 + 0.0215i)14-s + (−0.504 − 0.632i)15-s + (0.822 + 0.560i)16-s + (1.25 + 1.16i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0403 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0403 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.659576 - 0.686739i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.659576 - 0.686739i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.826 + 0.563i)T \) |
| 7 | \( 1 + (2.39 + 1.13i)T \) |
| good | 2 | \( 1 + (0.0539 + 0.00813i)T + (1.91 + 0.589i)T^{2} \) |
| 5 | \( 1 + (0.234 + 3.12i)T + (-4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (-1.33 - 3.39i)T + (-8.06 + 7.48i)T^{2} \) |
| 13 | \( 1 + (-2.60 + 3.26i)T + (-2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (-5.16 - 4.79i)T + (1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (2.58 + 4.47i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.48 + 1.37i)T + (1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (-0.417 + 1.82i)T + (-26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (1.02 - 1.77i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.73 + 0.844i)T + (30.5 - 20.8i)T^{2} \) |
| 41 | \( 1 + (-1.59 - 0.769i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (10.7 - 5.15i)T + (26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (-10.6 - 1.60i)T + (44.9 + 13.8i)T^{2} \) |
| 53 | \( 1 + (7.68 + 2.36i)T + (43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (0.511 - 6.82i)T + (-58.3 - 8.79i)T^{2} \) |
| 61 | \( 1 + (1.15 - 0.355i)T + (50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (-4.30 + 7.45i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.152 + 0.668i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-8.72 + 1.31i)T + (69.7 - 21.5i)T^{2} \) |
| 79 | \( 1 + (2.75 + 4.77i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.62 - 3.29i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (-3.75 + 9.57i)T + (-65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 + 8.70T + 97T^{2} \) |
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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
−12.84539783909224363018770833270, −12.42223723623254295846510627387, −10.43260009999244248360371228959, −9.515605892891877829255364535776, −8.730615503559145570203149085147, −7.82347122397288448276540744969, −6.18122458594241661037422955113, −4.81080740551010914371407085999, −3.70112985219481205978098791231, −1.05484780113664238924081801875,
3.11126765420360863587604739259, 3.77596393528022659517464613847, 5.73326100666578745437220593285, 6.94279038367837674720829614880, 8.263874363623835827496186588085, 9.272949960768623368144634598126, 10.05014856403161582031971848520, 11.22972770097927391242964875094, 12.34894808193326229620110266339, 13.72967496083942734257587508000
