| L(s) = 1 | + 99.9i·3-s − 610·5-s − 1.39e3i·7-s − 3.42e3·9-s − 1.84e4i·11-s + 5.47e3·13-s − 6.09e4i·15-s + 7.30e4·17-s − 1.94e4i·19-s + 1.39e5·21-s + 2.37e5i·23-s − 1.85e4·25-s + 3.13e5i·27-s + 1.28e5·29-s + 6.79e4i·31-s + ⋯ |
| L(s) = 1 | + 1.23i·3-s − 0.976·5-s − 0.582i·7-s − 0.521·9-s − 1.26i·11-s + 0.191·13-s − 1.20i·15-s + 0.875·17-s − 0.149i·19-s + 0.718·21-s + 0.847i·23-s − 0.0474·25-s + 0.589i·27-s + 0.181·29-s + 0.0735i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.47545\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.47545\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 99.9iT - 6.56e3T^{2} \) |
| 5 | \( 1 + 610T + 3.90e5T^{2} \) |
| 7 | \( 1 + 1.39e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + 1.84e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 5.47e3T + 8.15e8T^{2} \) |
| 17 | \( 1 - 7.30e4T + 6.97e9T^{2} \) |
| 19 | \( 1 + 1.94e4iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 2.37e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 1.28e5T + 5.00e11T^{2} \) |
| 31 | \( 1 - 6.79e4iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 3.47e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 2.14e6T + 7.98e12T^{2} \) |
| 43 | \( 1 + 5.92e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + 7.62e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 8.24e5T + 6.22e13T^{2} \) |
| 59 | \( 1 + 3.72e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 1.47e7T + 1.91e14T^{2} \) |
| 67 | \( 1 - 1.52e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 - 1.19e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 5.72e6T + 8.06e14T^{2} \) |
| 79 | \( 1 + 3.59e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 + 5.19e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 8.33e7T + 3.93e15T^{2} \) |
| 97 | \( 1 - 1.20e8T + 7.83e15T^{2} \) |
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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
−13.37616462565233790914174864650, −11.75958510373183490993354768687, −10.92897962510779414294763752497, −9.923664093719332757652865314180, −8.633702945814849767089338009083, −7.44751286963590300316291945565, −5.59513837838510094416668767674, −4.12442295323724367270428988216, −3.38750645087186340246926491883, −0.63307275723312919305251802875,
1.03321725869978853116927577304, 2.50993112200588389860417846615, 4.37337893284173940252729839676, 6.17231524327664394917947694639, 7.43448164004723637703366343760, 8.101762591965476280994931162787, 9.702490276936449846144991313921, 11.37172840882685673283418194405, 12.35772315185534093356642805183, 12.80762194927807919640080796236
