L(s) = 1 | − 11.3i·2-s − 128.·4-s − 233. i·5-s − 3.53e3·7-s + 1.44e3i·8-s − 2.64e3·10-s + 2.01e4i·11-s − 4.18e4·13-s + 3.99e4i·14-s + 1.63e4·16-s − 9.47e4i·17-s − 3.63e4·19-s + 2.98e4i·20-s + 2.28e5·22-s − 4.13e5i·23-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.500·4-s − 0.373i·5-s − 1.47·7-s + 0.353i·8-s − 0.264·10-s + 1.37i·11-s − 1.46·13-s + 1.04i·14-s + 0.250·16-s − 1.13i·17-s − 0.278·19-s + 0.186i·20-s + 0.974·22-s − 1.47i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.0590735 + 0.185861i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0590735 + 0.185861i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 11.3iT \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 233. iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 3.53e3T + 5.76e6T^{2} \) |
| 11 | \( 1 - 2.01e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 4.18e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 9.47e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 3.63e4T + 1.69e10T^{2} \) |
| 23 | \( 1 + 4.13e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 2.69e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 4.71e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + 3.00e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 1.71e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 3.62e6T + 1.16e13T^{2} \) |
| 47 | \( 1 - 6.01e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 1.02e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 2.68e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 5.44e6T + 1.91e14T^{2} \) |
| 67 | \( 1 + 6.12e6T + 4.06e14T^{2} \) |
| 71 | \( 1 - 2.11e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 4.90e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 8.35e6T + 1.51e15T^{2} \) |
| 83 | \( 1 - 5.13e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 1.07e8iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 2.04e7T + 7.83e15T^{2} \) |
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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
−16.14968740601839567274559786637, −14.57023672330232238167464165491, −12.84130577236346837842314280934, −12.24558213824575401240091870920, −10.17015811092955557983789660745, −9.280584036770352313080083503908, −7.02064824907503988817208373729, −4.70653189977563922506464077158, −2.61585276472302449532125512482, −0.098357000076130454853808495150,
3.35859632618951633508319508725, 5.80019996508245135146267048644, 7.15005099945856710634261426168, 8.999896508191296451017143115245, 10.42988872452461900709699907837, 12.48095525625212537133604321508, 13.73505940951187823737649356337, 15.10236266723732315934257461573, 16.30200561471739735573125992095, 17.25949815641182139764398727204