| L(s) = 1 | + (2.82 − 4.89i)2-s + (−15.9 − 27.7i)4-s + (−111. − 64.5i)5-s + (298. + 168. i)7-s − 181.·8-s + (−632. + 365. i)10-s + (1.18e3 + 2.05e3i)11-s + 820. i·13-s + (1.67e3 − 986. i)14-s + (−512. + 886. i)16-s + (2.28e3 − 1.31e3i)17-s + (−4.81e3 − 2.77e3i)19-s + 4.13e3i·20-s + 1.34e4·22-s + (5.69e3 − 9.86e3i)23-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.894 − 0.516i)5-s + (0.870 + 0.491i)7-s − 0.353·8-s + (−0.632 + 0.365i)10-s + (0.893 + 1.54i)11-s + 0.373i·13-s + (0.608 − 0.359i)14-s + (−0.125 + 0.216i)16-s + (0.464 − 0.268i)17-s + (−0.701 − 0.405i)19-s + 0.516i·20-s + 1.26·22-s + (0.468 − 0.810i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 + 0.597i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.801 + 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.241695612\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.241695612\) |
| \(L(4)\) |
|
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 | \( 1 + (-298. - 168. i)T \) |
| good | 5 | \( 1 + (111. + 64.5i)T + (7.81e3 + 1.35e4i)T^{2} \) |
| 11 | \( 1 + (-1.18e3 - 2.05e3i)T + (-8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 - 820. iT - 4.82e6T^{2} \) |
| 17 | \( 1 + (-2.28e3 + 1.31e3i)T + (1.20e7 - 2.09e7i)T^{2} \) |
| 19 | \( 1 + (4.81e3 + 2.77e3i)T + (2.35e7 + 4.07e7i)T^{2} \) |
| 23 | \( 1 + (-5.69e3 + 9.86e3i)T + (-7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 - 3.28e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + (-4.06e4 + 2.34e4i)T + (4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 + (1.08e4 - 1.88e4i)T + (-1.28e9 - 2.22e9i)T^{2} \) |
| 41 | \( 1 - 8.58e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.13e5T + 6.32e9T^{2} \) |
| 47 | \( 1 + (3.33e4 + 1.92e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 + (-1.66e4 - 2.87e4i)T + (-1.10e10 + 1.91e10i)T^{2} \) |
| 59 | \( 1 + (-2.89e5 + 1.67e5i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (1.45e5 + 8.42e4i)T + (2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-1.61e5 - 2.79e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 + 3.25e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + (-9.34e4 + 5.39e4i)T + (7.56e10 - 1.31e11i)T^{2} \) |
| 79 | \( 1 + (-1.69e5 + 2.93e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 - 2.24e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (3.31e5 + 1.91e5i)T + (2.48e11 + 4.30e11i)T^{2} \) |
| 97 | \( 1 - 3.26e4iT - 8.32e11T^{2} \) |
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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
−12.00620147640226904123424234589, −11.51782474974166436684537290394, −10.12879998784457534428974461286, −8.978055509592145376827943436011, −7.983831961171362677994872179391, −6.57343588785011130108831319439, −4.73498374755017369020209122095, −4.31867397292982440018836532625, −2.39845663157742689114621717263, −1.00802328929842101337806419978,
0.889206204896154774329870802089, 3.27480214183260080501061626121, 4.21177584064830967784401313026, 5.70019770618330198411903536876, 6.90951229549827929870433315630, 7.979076366795739041542347789028, 8.694008440378868089751086221598, 10.52466265760648675004449539378, 11.38085017497097305565802769737, 12.23443620634502081773857055142
