| L(s) = 1 | + 4·2-s + 16·4-s + 241.·7-s + 64·8-s − 653.·11-s − 828.·13-s + 966.·14-s + 256·16-s − 2.16e3·17-s + 1.25e3·19-s − 2.61e3·22-s − 3.74e3·23-s − 3.31e3·26-s + 3.86e3·28-s − 2.46e3·29-s − 1.89e3·31-s + 1.02e3·32-s − 8.64e3·34-s − 1.05e3·37-s + 5.01e3·38-s − 1.96e3·41-s − 1.10e4·43-s − 1.04e4·44-s − 1.49e4·46-s + 2.30e4·47-s + 4.15e4·49-s − 1.32e4·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.86·7-s + 0.353·8-s − 1.62·11-s − 1.35·13-s + 1.31·14-s + 0.250·16-s − 1.81·17-s + 0.797·19-s − 1.15·22-s − 1.47·23-s − 0.961·26-s + 0.931·28-s − 0.544·29-s − 0.354·31-s + 0.176·32-s − 1.28·34-s − 0.126·37-s + 0.563·38-s − 0.182·41-s − 0.908·43-s − 0.813·44-s − 1.04·46-s + 1.52·47-s + 2.47·49-s − 0.679·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 241.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 653.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 828.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.16e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.25e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.74e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.46e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.89e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.05e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.96e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.10e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.30e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.73e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.52e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.68e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.81e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.72e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.54e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.50e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.25e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.79e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.59e4T + 8.58e9T^{2} \) |
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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
−10.14899779328195236069171196220, −8.734606535378019109346665991985, −7.75698230636487782360478728166, −7.28380768610713417135931686922, −5.67324472543686894398659633686, −4.95871698913845500016925936815, −4.29500523162034436698643985115, −2.54115146327264989585544592697, −1.88751517892199282527874122681, 0,
1.88751517892199282527874122681, 2.54115146327264989585544592697, 4.29500523162034436698643985115, 4.95871698913845500016925936815, 5.67324472543686894398659633686, 7.28380768610713417135931686922, 7.75698230636487782360478728166, 8.734606535378019109346665991985, 10.14899779328195236069171196220
