| L(s) = 1 | − 0.290·2-s − 31.9·4-s − 87.1·5-s + 167.·7-s + 18.5·8-s + 25.2·10-s − 252.·11-s − 289.·13-s − 48.6·14-s + 1.01e3·16-s + 227.·17-s − 1.16e3·19-s + 2.78e3·20-s + 73.3·22-s + 1.23e3·23-s + 4.46e3·25-s + 83.9·26-s − 5.34e3·28-s − 1.97e3·29-s + 7.79e3·31-s − 888.·32-s − 66.0·34-s − 1.45e4·35-s + 7.04e3·37-s + 337.·38-s − 1.61e3·40-s + 1.12e4·41-s + ⋯ |
| L(s) = 1 | − 0.0512·2-s − 0.997·4-s − 1.55·5-s + 1.29·7-s + 0.102·8-s + 0.0799·10-s − 0.629·11-s − 0.474·13-s − 0.0662·14-s + 0.992·16-s + 0.190·17-s − 0.738·19-s + 1.55·20-s + 0.0323·22-s + 0.485·23-s + 1.42·25-s + 0.0243·26-s − 1.28·28-s − 0.436·29-s + 1.45·31-s − 0.153·32-s − 0.00979·34-s − 2.01·35-s + 0.846·37-s + 0.0378·38-s − 0.159·40-s + 1.04·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 59 | \( 1 - 3.48e3T \) |
| good | 2 | \( 1 + 0.290T + 32T^{2} \) |
| 5 | \( 1 + 87.1T + 3.12e3T^{2} \) |
| 7 | \( 1 - 167.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 252.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 289.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 227.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.16e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.23e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.97e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.79e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.04e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.12e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.85e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.01e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.73e4T + 4.18e8T^{2} \) |
| 61 | \( 1 + 998.T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.96e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.16e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.62e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.50e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.54e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.05e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.09e5T + 8.58e9T^{2} \) |
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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
−9.487260310552506616437280253843, −8.420872060903026312620766217756, −7.988536091709854798540370951363, −7.31970192761850733180645689252, −5.62473746941829503503219095790, −4.49158535815168912254335794121, −4.27689684902541750773050951459, −2.79705537698350487369587978975, −1.04222603964101307501305782262, 0,
1.04222603964101307501305782262, 2.79705537698350487369587978975, 4.27689684902541750773050951459, 4.49158535815168912254335794121, 5.62473746941829503503219095790, 7.31970192761850733180645689252, 7.988536091709854798540370951363, 8.420872060903026312620766217756, 9.487260310552506616437280253843
