| L(s) = 1 | − 24.1i·2-s − 325.·4-s − 436. i·5-s + 550.·7-s + 1.68e3i·8-s − 1.05e4·10-s + 2.87e4i·11-s − 7.92e3·13-s − 1.32e4i·14-s − 4.28e4·16-s + 3.52e4i·17-s − 1.98e5·19-s + 1.42e5i·20-s + 6.94e5·22-s + 3.98e5i·23-s + ⋯ |
| L(s) = 1 | − 1.50i·2-s − 1.27·4-s − 0.698i·5-s + 0.229·7-s + 0.410i·8-s − 1.05·10-s + 1.96i·11-s − 0.277·13-s − 0.345i·14-s − 0.653·16-s + 0.422i·17-s − 1.52·19-s + 0.888i·20-s + 2.96·22-s + 1.42i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.975843\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.975843\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 24.1iT - 256T^{2} \) |
| 5 | \( 1 + 436. iT - 3.90e5T^{2} \) |
| 7 | \( 1 - 550.T + 5.76e6T^{2} \) |
| 11 | \( 1 - 2.87e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 7.92e3T + 8.15e8T^{2} \) |
| 17 | \( 1 - 3.52e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 1.98e5T + 1.69e10T^{2} \) |
| 23 | \( 1 - 3.98e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 3.65e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 1.19e6T + 8.52e11T^{2} \) |
| 37 | \( 1 + 1.15e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + 3.49e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 8.06e5T + 1.16e13T^{2} \) |
| 47 | \( 1 - 3.75e5iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 8.04e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 1.29e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 2.64e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 1.85e7T + 4.06e14T^{2} \) |
| 71 | \( 1 - 2.02e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 3.11e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 3.57e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 7.32e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 3.07e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 1.05e8T + 7.83e15T^{2} \) |
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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
−12.54801512110854963884387683201, −11.78506768165182029221661732116, −10.49752538043445932126185665442, −9.716204875793924057661713149591, −8.614324805208230389910522815127, −7.01782031732466013214499033259, −4.94098936763936922810550464048, −4.03434380708416168750819311024, −2.28084239613971697004161613113, −1.36388049662529652995797823646,
0.30085181607075040020605247170, 2.79402706822446869477119678101, 4.59812011789456020445483376375, 6.05110141884479586968016199402, 6.67426638535912994692840983175, 8.101070682883867936044174390628, 8.746689049379067279908748018119, 10.50531246411774329622632586099, 11.47564463602374936805783081699, 13.19294563976180685298732478089
