| L(s) = 1 | + (13.5 + 7.79i)3-s + (−151. + 87.2i)5-s + (−180. − 291. i)7-s + (121.5 + 210. i)9-s + (642. − 1.11e3i)11-s + 160. i·13-s − 2.71e3·15-s + (3.05e3 + 1.76e3i)17-s + (586. − 338. i)19-s + (−158. − 5.34e3i)21-s + (−7.60e3 − 1.31e4i)23-s + (7.40e3 − 1.28e4i)25-s + 3.78e3i·27-s − 164.·29-s + (3.84e4 + 2.21e4i)31-s + ⋯ |
| L(s) = 1 | + (0.5 + 0.288i)3-s + (−1.20 + 0.697i)5-s + (−0.525 − 0.850i)7-s + (0.166 + 0.288i)9-s + (0.482 − 0.836i)11-s + 0.0731i·13-s − 0.805·15-s + (0.622 + 0.359i)17-s + (0.0855 − 0.0493i)19-s + (−0.0171 − 0.577i)21-s + (−0.624 − 1.08i)23-s + (0.473 − 0.820i)25-s + 0.192i·27-s − 0.00676·29-s + (1.28 + 0.744i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.155 - 0.987i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.155 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.212639666\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.212639666\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-13.5 - 7.79i)T \) |
| 7 | \( 1 + (180. + 291. i)T \) |
| good | 5 | \( 1 + (151. - 87.2i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 11 | \( 1 + (-642. + 1.11e3i)T + (-8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 - 160. iT - 4.82e6T^{2} \) |
| 17 | \( 1 + (-3.05e3 - 1.76e3i)T + (1.20e7 + 2.09e7i)T^{2} \) |
| 19 | \( 1 + (-586. + 338. i)T + (2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (7.60e3 + 1.31e4i)T + (-7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 + 164.T + 5.94e8T^{2} \) |
| 31 | \( 1 + (-3.84e4 - 2.21e4i)T + (4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + (3.16e4 + 5.48e4i)T + (-1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 - 6.87e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.16e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + (3.43e4 - 1.98e4i)T + (5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 + (6.07e4 - 1.05e5i)T + (-1.10e10 - 1.91e10i)T^{2} \) |
| 59 | \( 1 + (-2.61e5 - 1.50e5i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (1.28e5 - 7.43e4i)T + (2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (2.72e5 - 4.71e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 - 2.69e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + (4.41e5 + 2.54e5i)T + (7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (1.53e5 + 2.65e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 - 9.55e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (-8.10e5 + 4.67e5i)T + (2.48e11 - 4.30e11i)T^{2} \) |
| 97 | \( 1 + 3.57e5iT - 8.32e11T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.68219146169339864810184685022, −10.06414376515172696881541030864, −8.774847371744304673767460939646, −7.949754039289105123445159395524, −7.10688672654501004331524878688, −6.16267761645069302587485940463, −4.40145124201292229529841276796, −3.65527539144488688862919984894, −2.88428761621896794679511150713, −0.934410275771490822911713391397,
0.33022704035389851629137418557, 1.72194472341132328889591422630, 3.12548572579432211705307772448, 4.09721022929795878717098859403, 5.23623399011769763672890696883, 6.55848791337691820391045313958, 7.64066537377350716226766986396, 8.310150848450488336557024873257, 9.265741648993601214264242091100, 9.990105986026701387049765558252
