| L(s) = 1 | + (−1.23 − 3.80i)2-s − 28.1i·3-s + (−12.9 + 9.40i)4-s + (68.7 − 49.9i)5-s + (−106. + 34.7i)6-s + (−165. − 53.6i)7-s + (51.7 + 37.6i)8-s − 547.·9-s + (−274. − 199. i)10-s + (310. − 427. i)11-s + (264. + 363. i)12-s + (828. − 269. i)13-s + 694. i·14-s + (−1.40e3 − 1.93e3i)15-s + (79.1 − 243. i)16-s + (−869. + 1.19e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 − 0.672i)2-s − 1.80i·3-s + (−0.404 + 0.293i)4-s + (1.22 − 0.893i)5-s + (−1.21 + 0.394i)6-s + (−1.27 − 0.413i)7-s + (0.286 + 0.207i)8-s − 2.25·9-s + (−0.869 − 0.631i)10-s + (0.773 − 1.06i)11-s + (0.529 + 0.729i)12-s + (1.35 − 0.441i)13-s + 0.947i·14-s + (−1.61 − 2.21i)15-s + (0.0772 − 0.237i)16-s + (−0.729 + 1.00i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.706 - 0.707i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.706 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.590089 + 1.42327i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.590089 + 1.42327i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.23 + 3.80i)T \) |
| 41 | \( 1 + (3.60e3 - 1.01e4i)T \) |
| good | 3 | \( 1 + 28.1iT - 243T^{2} \) |
| 5 | \( 1 + (-68.7 + 49.9i)T + (965. - 2.97e3i)T^{2} \) |
| 7 | \( 1 + (165. + 53.6i)T + (1.35e4 + 9.87e3i)T^{2} \) |
| 11 | \( 1 + (-310. + 427. i)T + (-4.97e4 - 1.53e5i)T^{2} \) |
| 13 | \( 1 + (-828. + 269. i)T + (3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (869. - 1.19e3i)T + (-4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-881. - 286. i)T + (2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + (-56.9 - 175. i)T + (-5.20e6 + 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-1.04e3 - 1.43e3i)T + (-6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-7.27e3 - 5.28e3i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-3.93e3 + 2.86e3i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 43 | \( 1 + (4.72e3 + 1.45e4i)T + (-1.18e8 + 8.64e7i)T^{2} \) |
| 47 | \( 1 + (-1.04e4 + 3.40e3i)T + (1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (8.19e3 + 1.12e4i)T + (-1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (6.46e3 + 1.99e4i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (1.32e4 - 4.07e4i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + (3.14e4 + 4.33e4i)T + (-4.17e8 + 1.28e9i)T^{2} \) |
| 71 | \( 1 + (-1.00e4 + 1.38e4i)T + (-5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + 3.70e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.29e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 2.59e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-1.20e4 - 3.91e3i)T + (4.51e9 + 3.28e9i)T^{2} \) |
| 97 | \( 1 + (-6.75e4 - 9.29e4i)T + (-2.65e9 + 8.16e9i)T^{2} \) |
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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
−12.92581975674398195041434456924, −11.92631833995553595705638872455, −10.53539331369389216060948414977, −9.059961766818924418192612325133, −8.370615807912991076696676711950, −6.51522398343939173528584673896, −5.93644737912880639369752052822, −3.23812451925493579484109749409, −1.53846995799640217450022885574, −0.75126849128302206001293185947,
2.87277822305599110709184878225, 4.36730595437854092860982389052, 5.91583030491097082743012234248, 6.62592214970570899106098129019, 9.053798034314138088548987710797, 9.567250062237572960392962771807, 10.19680754924034129046921373766, 11.45611942918886671650951999522, 13.48847790484039413549689676020, 14.25395947971451173721499972447
