| L(s) = 1 | − 8.27·2-s + 36.4·4-s + 28.7·5-s − 37.0·8-s − 237.·10-s + 270.·11-s − 300.·13-s − 860.·16-s + 613.·17-s + 1.70e3·19-s + 1.04e3·20-s − 2.23e3·22-s − 3.18e3·23-s − 2.29e3·25-s + 2.48e3·26-s − 4.29e3·29-s − 2.02e3·31-s + 8.30e3·32-s − 5.07e3·34-s + 5.15e3·37-s − 1.40e4·38-s − 1.06e3·40-s − 7.14e3·41-s − 1.95e4·43-s + 9.85e3·44-s + 2.63e4·46-s + 1.99e4·47-s + ⋯ |
| L(s) = 1 | − 1.46·2-s + 1.13·4-s + 0.514·5-s − 0.204·8-s − 0.752·10-s + 0.673·11-s − 0.493·13-s − 0.840·16-s + 0.514·17-s + 1.08·19-s + 0.586·20-s − 0.984·22-s − 1.25·23-s − 0.735·25-s + 0.721·26-s − 0.949·29-s − 0.379·31-s + 1.43·32-s − 0.752·34-s + 0.618·37-s − 1.58·38-s − 0.105·40-s − 0.663·41-s − 1.61·43-s + 0.767·44-s + 1.83·46-s + 1.32·47-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
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 8.27T + 32T^{2} \) |
| 5 | \( 1 - 28.7T + 3.12e3T^{2} \) |
| 11 | \( 1 - 270.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 300.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 613.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.70e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.18e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.29e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.02e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.15e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.14e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.95e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.99e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.94e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.97e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.05e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.05e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.28e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.11e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.18e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.18e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.16e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.36e4T + 8.58e9T^{2} \) |
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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
−9.697428841727745129807689366502, −9.214452573120888325537701759667, −8.081291166741266461969409835341, −7.41705278068263277243976127402, −6.37213224899607405096134426003, −5.26214539941958616225140420240, −3.77376393854909991999598340414, −2.19987874558718285690459709306, −1.27421769244369166549244776652, 0,
1.27421769244369166549244776652, 2.19987874558718285690459709306, 3.77376393854909991999598340414, 5.26214539941958616225140420240, 6.37213224899607405096134426003, 7.41705278068263277243976127402, 8.081291166741266461969409835341, 9.214452573120888325537701759667, 9.697428841727745129807689366502
