L(s) = 1 | − 1.77i·2-s + 0.848·4-s + 2.98i·5-s + 6.83i·7-s − 8.60i·8-s + 5.30·10-s − 6.88·11-s + 12.4·13-s + 12.1·14-s − 11.8·16-s + 15.8·17-s + 19.2i·19-s + 2.53i·20-s + 12.2i·22-s + 33.2·23-s + ⋯ |
L(s) = 1 | − 0.887i·2-s + 0.212·4-s + 0.597i·5-s + 0.975i·7-s − 1.07i·8-s + 0.530·10-s − 0.626·11-s + 0.953·13-s + 0.866·14-s − 0.742·16-s + 0.930·17-s + 1.01i·19-s + 0.126i·20-s + 0.555i·22-s + 1.44·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.309i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.950 + 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.97444 - 0.313623i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.97444 - 0.313623i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 43 | \( 1 + (40.8 + 13.3i)T \) |
good | 2 | \( 1 + 1.77iT - 4T^{2} \) |
| 5 | \( 1 - 2.98iT - 25T^{2} \) |
| 7 | \( 1 - 6.83iT - 49T^{2} \) |
| 11 | \( 1 + 6.88T + 121T^{2} \) |
| 13 | \( 1 - 12.4T + 169T^{2} \) |
| 17 | \( 1 - 15.8T + 289T^{2} \) |
| 19 | \( 1 - 19.2iT - 361T^{2} \) |
| 23 | \( 1 - 33.2T + 529T^{2} \) |
| 29 | \( 1 - 45.8iT - 841T^{2} \) |
| 31 | \( 1 - 14.7T + 961T^{2} \) |
| 37 | \( 1 + 13.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 2.49T + 1.68e3T^{2} \) |
| 47 | \( 1 - 10.2T + 2.20e3T^{2} \) |
| 53 | \( 1 - 31.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + 64.4T + 3.48e3T^{2} \) |
| 61 | \( 1 + 78.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 89.3T + 4.48e3T^{2} \) |
| 71 | \( 1 + 35.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 35.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 50.1T + 6.24e3T^{2} \) |
| 83 | \( 1 - 10.3T + 6.88e3T^{2} \) |
| 89 | \( 1 - 13.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 66.6T + 9.40e3T^{2} \) |
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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
−10.86064586626415100461040846379, −10.57279063903127635837657342697, −9.427597942368995482788050969298, −8.453154358708608097264367166040, −7.25702963161859734902086234409, −6.27093037223253252034215344935, −5.23193147577659557130346487240, −3.47515946497530942431725075244, −2.80469930528354650983576661720, −1.41997362189293105493534158016,
1.02725372611106603818606312251, 2.92832774457165736222595715221, 4.49530264427287705658580580723, 5.43332345183213842953309781284, 6.51169968243079080810465839365, 7.38663409442561523311221014013, 8.163675514498740789968109982800, 9.051892680171223529907555234358, 10.35251979536991864039869886143, 11.06603053269841841128669323773