| L(s) = 1 | + 10.0i·2-s + 155.·4-s − 594. i·5-s + 4.40e3·7-s + 4.12e3i·8-s + 5.95e3·10-s − 3.47e3i·11-s − 1.82e4·13-s + 4.41e4i·14-s − 1.43e3·16-s − 8.66e4i·17-s + 7.61e4·19-s − 9.25e4i·20-s + 3.48e4·22-s − 5.24e5i·23-s + ⋯ |
| L(s) = 1 | + 0.625i·2-s + 0.608·4-s − 0.951i·5-s + 1.83·7-s + 1.00i·8-s + 0.595·10-s − 0.237i·11-s − 0.639·13-s + 1.14i·14-s − 0.0219·16-s − 1.03i·17-s + 0.584·19-s − 0.578i·20-s + 0.148·22-s − 1.87i·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
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(3.01794\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.01794\) |
| \(L(5)\) |
|
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 | \( 1 - 10.0iT - 256T^{2} \) |
| 5 | \( 1 + 594. iT - 3.90e5T^{2} \) |
| 7 | \( 1 - 4.40e3T + 5.76e6T^{2} \) |
| 11 | \( 1 + 3.47e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 1.82e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 8.66e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 7.61e4T + 1.69e10T^{2} \) |
| 23 | \( 1 + 5.24e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 7.75e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 9.25e5T + 8.52e11T^{2} \) |
| 37 | \( 1 - 2.37e5T + 3.51e12T^{2} \) |
| 41 | \( 1 + 5.15e5iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 3.95e6T + 1.16e13T^{2} \) |
| 47 | \( 1 - 5.32e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 1.30e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 1.25e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 1.18e7T + 1.91e14T^{2} \) |
| 67 | \( 1 - 1.35e7T + 4.06e14T^{2} \) |
| 71 | \( 1 - 1.33e5iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 5.24e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 4.78e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 5.85e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 4.62e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 8.82e7T + 7.83e15T^{2} \) |
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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
−12.56240536412800795661568944957, −11.61490113396439936864533510536, −10.75477461137441403380815673176, −8.935098858714048989839259822645, −8.063774592856295511154944835558, −7.10230145411722164736830188994, −5.37011639998131330909383829471, −4.73430123110603231920620898444, −2.34218704204383783259940322548, −1.01306024883059006585884642418,
1.41278188214716657411420528501, 2.37487306616179628110119600447, 3.88199410887090399882976070893, 5.53136367445432810389645802366, 7.13392133013255365515851429582, 7.910382129713072628070712169899, 9.723571527827451813811409077002, 10.90006266950514998736240883591, 11.34106602879612916972524099901, 12.34763654059015081563097148419
