| L(s) = 1 | − 5.29·2-s − 3.99·4-s − 37.0·5-s + 190.·8-s + 196.·10-s + 227.·11-s + 518·13-s − 880.·16-s + 777.·17-s + 1.48e3·19-s + 148.·20-s − 1.20e3·22-s − 3.82e3·23-s − 1.75e3·25-s − 2.74e3·26-s − 105.·29-s + 2.60e3·31-s − 1.43e3·32-s − 4.11e3·34-s + 402·37-s − 7.85e3·38-s − 7.05e3·40-s − 629.·41-s + 6.95e3·43-s − 910.·44-s + 2.02e4·46-s − 2.73e4·47-s + ⋯ |
| L(s) = 1 | − 0.935·2-s − 0.124·4-s − 0.662·5-s + 1.05·8-s + 0.619·10-s + 0.566·11-s + 0.850·13-s − 0.859·16-s + 0.652·17-s + 0.943·19-s + 0.0828·20-s − 0.530·22-s − 1.50·23-s − 0.560·25-s − 0.795·26-s − 0.0233·29-s + 0.486·31-s − 0.248·32-s − 0.610·34-s + 0.0482·37-s − 0.882·38-s − 0.697·40-s − 0.0585·41-s + 0.573·43-s − 0.0708·44-s + 1.41·46-s − 1.80·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9175029278\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9175029278\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 5.29T + 32T^{2} \) |
| 5 | \( 1 + 37.0T + 3.12e3T^{2} \) |
| 11 | \( 1 - 227.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 518T + 3.71e5T^{2} \) |
| 17 | \( 1 - 777.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.48e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.82e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 105.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.60e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 402T + 6.93e7T^{2} \) |
| 41 | \( 1 + 629.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.95e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.73e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.05e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.51e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.26e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.31e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.82e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.28e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.11e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.66e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.26e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.06e4T + 8.58e9T^{2} \) |
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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
−10.00116247716074189059479188803, −9.526474800273534871201817923813, −8.305255077742403388827073457113, −7.982003572835176557399778459608, −6.86915920766509742254908661931, −5.64640386277016770790403112806, −4.32456347513709346837766282755, −3.48591876297525799087459765635, −1.65592222135377935651714962993, −0.60539608153253436928732458693,
0.60539608153253436928732458693, 1.65592222135377935651714962993, 3.48591876297525799087459765635, 4.32456347513709346837766282755, 5.64640386277016770790403112806, 6.86915920766509742254908661931, 7.982003572835176557399778459608, 8.305255077742403388827073457113, 9.526474800273534871201817923813, 10.00116247716074189059479188803
