| L(s) = 1 | + (−0.247 + 1.98i)2-s + (−3.87 − 0.980i)4-s + 2.20·5-s + 8.35i·7-s + (2.90 − 7.45i)8-s + (−0.544 + 4.36i)10-s + 5.25i·11-s + 14.7·13-s + (−16.5 − 2.06i)14-s + (14.0 + 7.60i)16-s − 28.2·17-s + 19.1i·19-s + (−8.53 − 2.15i)20-s + (−10.4 − 1.29i)22-s + 3.65i·23-s + ⋯ |
| L(s) = 1 | + (−0.123 + 0.992i)2-s + (−0.969 − 0.245i)4-s + 0.440·5-s + 1.19i·7-s + (0.363 − 0.931i)8-s + (−0.0544 + 0.436i)10-s + 0.477i·11-s + 1.13·13-s + (−1.18 − 0.147i)14-s + (0.879 + 0.475i)16-s − 1.66·17-s + 1.00i·19-s + (−0.426 − 0.107i)20-s + (−0.473 − 0.0589i)22-s + 0.158i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.245i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.969 - 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.143568 + 1.15311i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.143568 + 1.15311i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.247 - 1.98i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2.20T + 25T^{2} \) |
| 7 | \( 1 - 8.35iT - 49T^{2} \) |
| 11 | \( 1 - 5.25iT - 121T^{2} \) |
| 13 | \( 1 - 14.7T + 169T^{2} \) |
| 17 | \( 1 + 28.2T + 289T^{2} \) |
| 19 | \( 1 - 19.1iT - 361T^{2} \) |
| 23 | \( 1 - 3.65iT - 529T^{2} \) |
| 29 | \( 1 + 24.6T + 841T^{2} \) |
| 31 | \( 1 - 38.0iT - 961T^{2} \) |
| 37 | \( 1 + 4.21T + 1.36e3T^{2} \) |
| 41 | \( 1 + 19.8T + 1.68e3T^{2} \) |
| 43 | \( 1 - 23.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 29.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 32.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 9.19iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 81.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + 7.92iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 62.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 33.3T + 5.32e3T^{2} \) |
| 79 | \( 1 - 62.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 118. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 107.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 3.57T + 9.40e3T^{2} \) |
| 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
−11.93659881615871689331671812927, −10.73764399595792858783097694254, −9.627811413847719094122409115803, −8.866785445678717237547036702020, −8.184395454608023990768176253246, −6.82151119036996468033211667889, −6.01460111353563917246543543632, −5.18844393464460216246613442863, −3.83056129483285368555665165013, −1.90118576404337006307312400210,
0.57757012348477534241470192425, 2.10807548130997069065800666121, 3.64515146761856458486679951048, 4.49106681535579156682925416301, 5.94049850072176071223788486119, 7.19030112410836185935263927544, 8.441331467309351312698621094827, 9.220989291101827410241671223225, 10.22716068454348049625094186062, 11.07141272215424190581227717610
