| L(s) = 1 | + (−2.82 − 4.89i)2-s + (−15.9 + 27.7i)4-s + (10.7 − 6.20i)5-s + (195. + 281. i)7-s + 181.·8-s + (−60.8 − 35.1i)10-s + (−774. + 1.34e3i)11-s − 2.77e3i·13-s + (826. − 1.75e3i)14-s + (−512. − 886. i)16-s + (109. + 63.2i)17-s + (−1.14e3 + 662. i)19-s + 397. i·20-s + 8.76e3·22-s + (−7.11e3 − 1.23e4i)23-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.0859 − 0.0496i)5-s + (0.570 + 0.821i)7-s + 0.353·8-s + (−0.0608 − 0.0351i)10-s + (−0.582 + 1.00i)11-s − 1.26i·13-s + (0.301 − 0.639i)14-s + (−0.125 − 0.216i)16-s + (0.0223 + 0.0128i)17-s + (−0.167 + 0.0966i)19-s + 0.0496i·20-s + 0.823·22-s + (−0.585 − 1.01i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.669 - 0.742i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.669 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.3548750569\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3548750569\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.82 + 4.89i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-195. - 281. i)T \) |
| good | 5 | \( 1 + (-10.7 + 6.20i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 11 | \( 1 + (774. - 1.34e3i)T + (-8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 + 2.77e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + (-109. - 63.2i)T + (1.20e7 + 2.09e7i)T^{2} \) |
| 19 | \( 1 + (1.14e3 - 662. i)T + (2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (7.11e3 + 1.23e4i)T + (-7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 - 7.47e3T + 5.94e8T^{2} \) |
| 31 | \( 1 + (4.92e4 + 2.84e4i)T + (4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + (-4.50e4 - 7.79e4i)T + (-1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + 3.57e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 7.94e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + (1.26e5 - 7.29e4i)T + (5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 + (8.50e4 - 1.47e5i)T + (-1.10e10 - 1.91e10i)T^{2} \) |
| 59 | \( 1 + (-1.87e5 - 1.08e5i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (1.72e5 - 9.95e4i)T + (2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (7.22e4 - 1.25e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 + 4.07e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + (1.61e5 + 9.33e4i)T + (7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (4.19e4 + 7.26e4i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + 1.62e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (-4.39e5 + 2.53e5i)T + (2.48e11 - 4.30e11i)T^{2} \) |
| 97 | \( 1 + 5.09e5iT - 8.32e11T^{2} \) |
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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
−12.54386543303486732401777206155, −11.56709272072817989247351613210, −10.49099897699755813871252016726, −9.609571778915267483145478277346, −8.390616800107316394739386940104, −7.56451249353094569733892675880, −5.76856965601031571844591924410, −4.60030011225941852852399668324, −2.87028493360759388365970612818, −1.70482437113873236516592926152,
0.12183744369318971583305144574, 1.72211884001606428282822782236, 3.81601396409375956926608745203, 5.16039313395119310366664114735, 6.44098241937775524058275560057, 7.55266310155825117146013999887, 8.485045447642040363926889082187, 9.648161490140671133806221630311, 10.77498692861956216557350945229, 11.58662552417194622896027107045
