L(s) = 1 | + (−1.76 − 3.04i)2-s + (−4.19 + 7.26i)4-s + (1.93 − 1.11i)5-s + (0.244 + 6.99i)7-s + 15.4·8-s + (−6.81 − 3.93i)10-s + (1.29 − 2.24i)11-s − 11.5i·13-s + (20.8 − 13.0i)14-s + (−10.4 − 18.0i)16-s + (−20.0 − 11.6i)17-s + (25.9 − 14.9i)19-s + 18.7i·20-s − 9.13·22-s + (−17.5 − 30.4i)23-s + ⋯ |
L(s) = 1 | + (−0.880 − 1.52i)2-s + (−1.04 + 1.81i)4-s + (0.387 − 0.223i)5-s + (0.0348 + 0.999i)7-s + 1.93·8-s + (−0.681 − 0.393i)10-s + (0.117 − 0.204i)11-s − 0.890i·13-s + (1.49 − 0.932i)14-s + (−0.652 − 1.12i)16-s + (−1.18 − 0.682i)17-s + (1.36 − 0.788i)19-s + 0.938i·20-s − 0.415·22-s + (−0.763 − 1.32i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.160i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.986 + 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0640377 - 0.790746i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0640377 - 0.790746i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-1.93 + 1.11i)T \) |
| 7 | \( 1 + (-0.244 - 6.99i)T \) |
good | 2 | \( 1 + (1.76 + 3.04i)T + (-2 + 3.46i)T^{2} \) |
| 11 | \( 1 + (-1.29 + 2.24i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 11.5iT - 169T^{2} \) |
| 17 | \( 1 + (20.0 + 11.6i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-25.9 + 14.9i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (17.5 + 30.4i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 - 24.4T + 841T^{2} \) |
| 31 | \( 1 + (32.4 + 18.7i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (12.8 + 22.2i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 3.71iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 74.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + (2.92 - 1.68i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (20.0 - 34.6i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-42.7 - 24.6i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (0.765 - 0.441i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-32.5 + 56.3i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 86.0T + 5.04e3T^{2} \) |
| 73 | \( 1 + (53.3 + 30.7i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (13.7 + 23.8i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 131. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-56.5 + 32.6i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 42.2iT - 9.40e3T^{2} \) |
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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
−10.99913214531092364986163042779, −10.10221399488983145472349529790, −9.124980020208181707493352940219, −8.769026124799091672626264402212, −7.56457569136438139614870459974, −5.92480538665788014842495128337, −4.61398680768214994549951007694, −3.01333675422428165578990140884, −2.20186481452888995731601463468, −0.53976035201160986702370566882,
1.45176051864671167295225043253, 3.96571254822751236280642356978, 5.29552624611802647416862450089, 6.37535961173381785044381537725, 7.11247628222619937651053690413, 7.86719537952701455455963121634, 8.982872574484194104637964974201, 9.753433998685507172162154506736, 10.50127613395885787435538874882, 11.65821610798888381363245758127