| L(s) = 1 | + (−501. + 100. i)2-s + (2.41e5 − 1.01e5i)4-s + 2.68e6i·5-s + 4.44e7i·7-s + (−1.11e8 + 7.51e7i)8-s + (−2.70e8 − 1.34e9i)10-s + 2.94e9·11-s − 2.11e10i·13-s + (−4.48e9 − 2.23e10i)14-s + (4.82e10 − 4.89e10i)16-s + 8.60e10·17-s − 1.59e11·19-s + (2.71e11 + 6.49e11i)20-s + (−1.47e12 + 2.96e11i)22-s + 8.84e11i·23-s + ⋯ |
| L(s) = 1 | + (−0.980 + 0.196i)2-s + (0.922 − 0.385i)4-s + 1.37i·5-s + 1.10i·7-s + (−0.828 + 0.560i)8-s + (−0.270 − 1.34i)10-s + 1.24·11-s − 1.99i·13-s + (−0.217 − 1.08i)14-s + (0.702 − 0.712i)16-s + 0.725·17-s − 0.493·19-s + (0.530 + 1.26i)20-s + (−1.22 + 0.245i)22-s + 0.491i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.828 + 0.560i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (-0.828 + 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{19}{2})\) |
\(\approx\) |
\(0.4456542969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4456542969\) |
| \(L(10)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (501. - 100. i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2.68e6iT - 3.81e12T^{2} \) |
| 7 | \( 1 - 4.44e7iT - 1.62e15T^{2} \) |
| 11 | \( 1 - 2.94e9T + 5.55e18T^{2} \) |
| 13 | \( 1 + 2.11e10iT - 1.12e20T^{2} \) |
| 17 | \( 1 - 8.60e10T + 1.40e22T^{2} \) |
| 19 | \( 1 + 1.59e11T + 1.04e23T^{2} \) |
| 23 | \( 1 - 8.84e11iT - 3.24e24T^{2} \) |
| 29 | \( 1 + 1.10e13iT - 2.10e26T^{2} \) |
| 31 | \( 1 - 9.18e12iT - 6.99e26T^{2} \) |
| 37 | \( 1 + 1.80e13iT - 1.68e28T^{2} \) |
| 41 | \( 1 + 4.66e14T + 1.07e29T^{2} \) |
| 43 | \( 1 - 1.02e14T + 2.52e29T^{2} \) |
| 47 | \( 1 - 1.49e15iT - 1.25e30T^{2} \) |
| 53 | \( 1 - 4.93e14iT - 1.08e31T^{2} \) |
| 59 | \( 1 + 1.08e16T + 7.50e31T^{2} \) |
| 61 | \( 1 + 7.51e15iT - 1.36e32T^{2} \) |
| 67 | \( 1 + 1.79e16T + 7.40e32T^{2} \) |
| 71 | \( 1 - 4.48e16iT - 2.10e33T^{2} \) |
| 73 | \( 1 + 5.61e16T + 3.46e33T^{2} \) |
| 79 | \( 1 - 2.04e17iT - 1.43e34T^{2} \) |
| 83 | \( 1 + 2.79e17T + 3.49e34T^{2} \) |
| 89 | \( 1 - 3.39e17T + 1.22e35T^{2} \) |
| 97 | \( 1 + 3.43e17T + 5.77e35T^{2} \) |
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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
−11.48857366206146581860026149683, −10.52837437489914742055939414032, −9.669311103422907748501918854406, −8.472158492382995321140245561021, −7.47819755406750210237318480044, −6.34223019107444261837941200554, −5.61763217613207165917879808677, −3.30516036338461169311918516698, −2.59592342591044789655727967627, −1.29185937809932612558339630013,
0.12666724063917966449770927810, 1.21150368892474617397649482292, 1.71665621520077398736907391874, 3.71809309184000579588363039511, 4.56618187478175052308382704906, 6.39375112377930003505365910814, 7.28274100301138625506068047949, 8.662990244287573602968358248882, 9.187742481505433799098788682331, 10.28167856992709600297839076572
