| L(s) = 1 | + (6.92 − 4i)2-s + (33.2 + 57.5i)3-s + (31.9 − 55.4i)4-s + 355. i·5-s + (460. + 265. i)6-s + (297. + 171.5i)7-s − 511. i·8-s + (−1.11e3 + 1.92e3i)9-s + (1.42e3 + 2.46e3i)10-s + (−1.63e3 + 942. i)11-s + 4.24e3·12-s + (−7.78e3 + 1.46e3i)13-s + 2.74e3·14-s + (−2.04e4 + 1.17e4i)15-s + (−2.04e3 − 3.54e3i)16-s + (−1.01e4 + 1.75e4i)17-s + ⋯ |
| L(s) = 1 | + (0.612 − 0.353i)2-s + (0.709 + 1.22i)3-s + (0.249 − 0.433i)4-s + 1.27i·5-s + (0.869 + 0.502i)6-s + (0.327 + 0.188i)7-s − 0.353i·8-s + (−0.508 + 0.880i)9-s + (0.449 + 0.778i)10-s + (−0.369 + 0.213i)11-s + 0.709·12-s + (−0.982 + 0.185i)13-s + 0.267·14-s + (−1.56 + 0.902i)15-s + (−0.125 − 0.216i)16-s + (−0.500 + 0.867i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.343i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.844673936\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.844673936\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-6.92 + 4i)T \) |
| 7 | \( 1 + (-297. - 171.5i)T \) |
| 13 | \( 1 + (7.78e3 - 1.46e3i)T \) |
| good | 3 | \( 1 + (-33.2 - 57.5i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 - 355. iT - 7.81e4T^{2} \) |
| 11 | \( 1 + (1.63e3 - 942. i)T + (9.74e6 - 1.68e7i)T^{2} \) |
| 17 | \( 1 + (1.01e4 - 1.75e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-1.36e4 - 7.86e3i)T + (4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (1.93e4 + 3.35e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (3.25e3 + 5.63e3i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 - 5.92e3iT - 2.75e10T^{2} \) |
| 37 | \( 1 + (2.15e5 - 1.24e5i)T + (4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-1.87e5 + 1.08e5i)T + (9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-1.39e5 + 2.42e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 - 2.02e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + 1.51e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-4.97e5 - 2.87e5i)T + (1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (4.83e5 - 8.38e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.99e6 + 1.14e6i)T + (3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-5.34e5 - 3.08e5i)T + (4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 + 9.19e4iT - 1.10e13T^{2} \) |
| 79 | \( 1 + 4.56e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.07e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + (3.76e6 - 2.17e6i)T + (2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + (1.30e6 + 7.54e5i)T + (4.03e13 + 6.99e13i)T^{2} \) |
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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
−11.60047192290200407890828810304, −10.53352370572497528703779577663, −10.17197740695592137370034512630, −9.043106568279628419806050392620, −7.70636935803808124438556555614, −6.46514089026793588384228271616, −5.04914453315241178252984720232, −4.04110434368717031295405543466, −3.02753895355673702569634366064, −2.16569966174364498274484102634,
0.48267311920405125438627930456, 1.72907772046454736189080293801, 2.89419466932939339155713837792, 4.57513167898591927572646190209, 5.45889164639359933338810800709, 6.97001067277924012670576114369, 7.72973665110100396898829748579, 8.512397514470814440778966562309, 9.554198051626726929378352460275, 11.35189342651930008268836753467
