| L(s) = 1 | − 8.54·2-s + 41.0·4-s + 25·5-s − 123.·7-s − 77.7·8-s − 213.·10-s − 121·11-s + 635.·13-s + 1.05e3·14-s − 650.·16-s − 706.·17-s − 203.·19-s + 1.02e3·20-s + 1.03e3·22-s − 2.05e3·23-s + 625·25-s − 5.43e3·26-s − 5.07e3·28-s − 3.84e3·29-s + 3.60e3·31-s + 8.04e3·32-s + 6.04e3·34-s − 3.08e3·35-s + 1.52e4·37-s + 1.74e3·38-s − 1.94e3·40-s + 1.88e4·41-s + ⋯ |
| L(s) = 1 | − 1.51·2-s + 1.28·4-s + 0.447·5-s − 0.951·7-s − 0.429·8-s − 0.675·10-s − 0.301·11-s + 1.04·13-s + 1.43·14-s − 0.634·16-s − 0.593·17-s − 0.129·19-s + 0.574·20-s + 0.455·22-s − 0.811·23-s + 0.200·25-s − 1.57·26-s − 1.22·28-s − 0.849·29-s + 0.673·31-s + 1.38·32-s + 0.896·34-s − 0.425·35-s + 1.82·37-s + 0.195·38-s − 0.192·40-s + 1.75·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.6889959406\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6889959406\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 + 8.54T + 32T^{2} \) |
| 7 | \( 1 + 123.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 635.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 706.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 203.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.05e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.84e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.60e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.52e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.88e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.58e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.92e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.04e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.25e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.42e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 7.00e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.68e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.42e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 9.21e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.31e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.88e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.26e4T + 8.58e9T^{2} \) |
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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
−9.776931661615300845942270875875, −9.475176892992307765951478578912, −8.467324581147029056698147100852, −7.72391418676437764987766253969, −6.56351792128898461195227971132, −5.98537970460484887762150507070, −4.31657848040039089290636638133, −2.88261987933243581241243723194, −1.72533600011526643073181814504, −0.52776070111687816042049455174,
0.52776070111687816042049455174, 1.72533600011526643073181814504, 2.88261987933243581241243723194, 4.31657848040039089290636638133, 5.98537970460484887762150507070, 6.56351792128898461195227971132, 7.72391418676437764987766253969, 8.467324581147029056698147100852, 9.475176892992307765951478578912, 9.776931661615300845942270875875
