L(s) = 1 | − 22.4·2-s + (−67.3 + 45i)3-s + 248·4-s + (1.51e3 − 1.01e3i)6-s + 1.75e3i·7-s + 179.·8-s + (2.51e3 − 6.06e3i)9-s − 6.95e3i·11-s + (−1.67e4 + 1.11e4i)12-s + 2.57e4i·13-s − 3.92e4i·14-s − 6.75e4·16-s − 7.48e4·17-s + (−5.63e4 + 1.36e5i)18-s − 1.89e4·19-s + ⋯ |
L(s) = 1 | − 1.40·2-s + (−0.831 + 0.555i)3-s + 0.968·4-s + (1.16 − 0.779i)6-s + 0.728i·7-s + 0.0438·8-s + (0.382 − 0.923i)9-s − 0.475i·11-s + (−0.805 + 0.538i)12-s + 0.900i·13-s − 1.02i·14-s − 1.03·16-s − 0.896·17-s + (−0.536 + 1.29i)18-s − 0.145·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 + 0.495i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.868 + 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.307347 - 0.0814514i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.307347 - 0.0814514i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (67.3 - 45i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 22.4T + 256T^{2} \) |
| 7 | \( 1 - 1.75e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + 6.95e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 2.57e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + 7.48e4T + 6.97e9T^{2} \) |
| 19 | \( 1 + 1.89e4T + 1.69e10T^{2} \) |
| 23 | \( 1 + 4.70e5T + 7.83e10T^{2} \) |
| 29 | \( 1 - 4.60e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 3.51e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + 1.33e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 + 1.87e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 3.52e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 - 4.08e6T + 2.38e13T^{2} \) |
| 53 | \( 1 + 6.60e6T + 6.22e13T^{2} \) |
| 59 | \( 1 - 1.37e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 7.53e5T + 1.91e14T^{2} \) |
| 67 | \( 1 + 2.26e6iT - 4.06e14T^{2} \) |
| 71 | \( 1 + 1.70e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 2.76e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 - 2.29e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 4.63e7T + 2.25e15T^{2} \) |
| 89 | \( 1 - 7.26e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 1.47e8iT - 7.83e15T^{2} \) |
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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
−12.28684290326499868337782766031, −11.31555044233810425050916193094, −10.45082110408543680576143095898, −9.327035222889611915298808287029, −8.627534557001692721348142002575, −7.03147092777359259468538312687, −5.80066777165658366929060707671, −4.21012150461891235477603698052, −1.94596074272930092986494057165, −0.28752246922820041721980186388,
0.70112756587193685475036515377, 1.98675788241854501046629837510, 4.51398480959655594650912759312, 6.25971987182531883830799790650, 7.41730225025252755695091241705, 8.192975350988981454903177889740, 9.801980895748280923476553100146, 10.55331569360994764921540357719, 11.50030047079983131337415013922, 12.82026859361561444989090400491