| L(s) = 1 | − 8.71·2-s + 44.0·4-s − 99.6·5-s − 104.·8-s + 868.·10-s − 374.·11-s + 868.·13-s − 495.·16-s + 1.09e3·17-s − 868.·19-s − 4.38e3·20-s + 3.26e3·22-s − 2.97e3·23-s + 6.81e3·25-s − 7.57e3·26-s − 4.51e3·29-s − 9.55e3·31-s + 7.67e3·32-s − 9.55e3·34-s − 5.46e3·37-s + 7.57e3·38-s + 1.04e4·40-s − 9.86e3·41-s + 1.25e4·43-s − 1.64e4·44-s + 2.59e4·46-s + 9.96e3·47-s + ⋯ |
| L(s) = 1 | − 1.54·2-s + 1.37·4-s − 1.78·5-s − 0.577·8-s + 2.74·10-s − 0.934·11-s + 1.42·13-s − 0.484·16-s + 0.920·17-s − 0.552·19-s − 2.45·20-s + 1.43·22-s − 1.17·23-s + 2.17·25-s − 2.19·26-s − 0.997·29-s − 1.78·31-s + 1.32·32-s − 1.41·34-s − 0.656·37-s + 0.851·38-s + 1.03·40-s − 0.916·41-s + 1.03·43-s − 1.28·44-s + 1.80·46-s + 0.658·47-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
| \(L(3)\) |
\(\approx\) |
\(0.2219844973\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2219844973\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 8.71T + 32T^{2} \) |
| 5 | \( 1 + 99.6T + 3.12e3T^{2} \) |
| 11 | \( 1 + 374.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 868.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.09e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 868.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.97e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.51e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.55e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.46e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 9.86e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.25e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 9.96e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.51e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.26e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.77e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.99e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.14e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.86e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.01e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.07e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.08e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.33e5T + 8.58e9T^{2} \) |
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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
−10.55386834660206064227043275041, −9.147601277011634480392056367543, −8.434675646195359268099417963973, −7.74462590608702901226574664592, −7.26956299575436218335375215301, −5.82922022001886276778251643061, −4.24142733142983482289497017073, −3.29261813417647266645000541963, −1.62074843303436833791400965166, −0.31257288650261891516641592546,
0.31257288650261891516641592546, 1.62074843303436833791400965166, 3.29261813417647266645000541963, 4.24142733142983482289497017073, 5.82922022001886276778251643061, 7.26956299575436218335375215301, 7.74462590608702901226574664592, 8.434675646195359268099417963973, 9.147601277011634480392056367543, 10.55386834660206064227043275041
