L(s) = 1 | − 2.74·2-s − 24.4·4-s − 58.3·5-s + 155.·8-s + 160.·10-s − 17.4·11-s + 889.·13-s + 356.·16-s − 1.02e3·17-s − 1.73e3·19-s + 1.42e3·20-s + 47.8·22-s − 3.93e3·23-s + 281.·25-s − 2.44e3·26-s − 5.63e3·29-s − 3.09e3·31-s − 5.94e3·32-s + 2.82e3·34-s + 5.02e3·37-s + 4.78e3·38-s − 9.05e3·40-s − 1.83e4·41-s − 1.63e3·43-s + 425.·44-s + 1.08e4·46-s + 9.60e3·47-s + ⋯ |
L(s) = 1 | − 0.485·2-s − 0.763·4-s − 1.04·5-s + 0.856·8-s + 0.507·10-s − 0.0434·11-s + 1.46·13-s + 0.347·16-s − 0.861·17-s − 1.10·19-s + 0.797·20-s + 0.0210·22-s − 1.55·23-s + 0.0901·25-s − 0.709·26-s − 1.24·29-s − 0.578·31-s − 1.02·32-s + 0.418·34-s + 0.603·37-s + 0.537·38-s − 0.894·40-s − 1.70·41-s − 0.134·43-s + 0.0331·44-s + 0.753·46-s + 0.634·47-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
\(L(3)\) |
\(\approx\) |
\(0.5009835237\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5009835237\) |
\(L(\frac{7}{2})\) |
|
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 | \( 1 + 2.74T + 32T^{2} \) |
| 5 | \( 1 + 58.3T + 3.12e3T^{2} \) |
| 11 | \( 1 + 17.4T + 1.61e5T^{2} \) |
| 13 | \( 1 - 889.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.02e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.73e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.93e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.63e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.09e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.02e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.83e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.63e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 9.60e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.32e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.60e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.28e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.70e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.59e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.93e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.84e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.58e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.65e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 7.59e4T + 8.58e9T^{2} \) |
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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
−10.35698240395337473760883098579, −9.220805991549512824169296684717, −8.417998782571534870749689347403, −7.949886587238886812849391663884, −6.73933817755199891664261293165, −5.54945822207082317598282328618, −4.11142540964067218662149398226, −3.82535613665763969579904239496, −1.84714680412975526335315305331, −0.39480709810407028211308373607,
0.39480709810407028211308373607, 1.84714680412975526335315305331, 3.82535613665763969579904239496, 4.11142540964067218662149398226, 5.54945822207082317598282328618, 6.73933817755199891664261293165, 7.949886587238886812849391663884, 8.417998782571534870749689347403, 9.220805991549512824169296684717, 10.35698240395337473760883098579