| L(s) = 1 | − 257.·2-s − 6.47e4·4-s − 3.90e5·5-s + 8.43e6·7-s + 5.04e7·8-s + 1.00e8·10-s + 1.27e9·11-s + 2.16e9·13-s − 2.17e9·14-s − 4.50e9·16-s − 3.94e10·17-s + 1.12e11·19-s + 2.52e10·20-s − 3.28e11·22-s − 5.33e11·23-s + 1.52e11·25-s − 5.58e11·26-s − 5.46e11·28-s + 1.70e12·29-s − 1.16e11·31-s − 5.44e12·32-s + 1.01e13·34-s − 3.29e12·35-s + 3.69e13·37-s − 2.88e13·38-s − 1.97e13·40-s − 6.01e13·41-s + ⋯ |
| L(s) = 1 | − 0.711·2-s − 0.493·4-s − 0.447·5-s + 0.553·7-s + 1.06·8-s + 0.318·10-s + 1.79·11-s + 0.737·13-s − 0.393·14-s − 0.262·16-s − 1.37·17-s + 1.51·19-s + 0.220·20-s − 1.27·22-s − 1.41·23-s + 0.200·25-s − 0.524·26-s − 0.273·28-s + 0.632·29-s − 0.0244·31-s − 0.876·32-s + 0.974·34-s − 0.247·35-s + 1.72·37-s − 1.07·38-s − 0.475·40-s − 1.17·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(1.383551519\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.383551519\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 3.90e5T \) |
| good | 2 | \( 1 + 257.T + 1.31e5T^{2} \) |
| 7 | \( 1 - 8.43e6T + 2.32e14T^{2} \) |
| 11 | \( 1 - 1.27e9T + 5.05e17T^{2} \) |
| 13 | \( 1 - 2.16e9T + 8.65e18T^{2} \) |
| 17 | \( 1 + 3.94e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 1.12e11T + 5.48e21T^{2} \) |
| 23 | \( 1 + 5.33e11T + 1.41e23T^{2} \) |
| 29 | \( 1 - 1.70e12T + 7.25e24T^{2} \) |
| 31 | \( 1 + 1.16e11T + 2.25e25T^{2} \) |
| 37 | \( 1 - 3.69e13T + 4.56e26T^{2} \) |
| 41 | \( 1 + 6.01e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 3.10e13T + 5.87e27T^{2} \) |
| 47 | \( 1 - 2.34e14T + 2.66e28T^{2} \) |
| 53 | \( 1 + 5.42e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 1.85e14T + 1.27e30T^{2} \) |
| 61 | \( 1 + 1.71e15T + 2.24e30T^{2} \) |
| 67 | \( 1 + 6.75e12T + 1.10e31T^{2} \) |
| 71 | \( 1 + 9.21e13T + 2.96e31T^{2} \) |
| 73 | \( 1 - 1.14e16T + 4.74e31T^{2} \) |
| 79 | \( 1 - 8.98e15T + 1.81e32T^{2} \) |
| 83 | \( 1 + 4.25e15T + 4.21e32T^{2} \) |
| 89 | \( 1 - 3.54e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 3.08e16T + 5.95e33T^{2} \) |
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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
−11.95171543941184700994078855858, −11.09209874347164853165236327521, −9.634621045610657920503732259741, −8.768915652114068914057593438294, −7.78038714602054218944307532701, −6.39603590133567386274366081438, −4.63651036786332712530676458522, −3.73988491678600761202849703155, −1.65078518855088952769486592003, −0.72095957173651519041039429038,
0.72095957173651519041039429038, 1.65078518855088952769486592003, 3.73988491678600761202849703155, 4.63651036786332712530676458522, 6.39603590133567386274366081438, 7.78038714602054218944307532701, 8.768915652114068914057593438294, 9.634621045610657920503732259741, 11.09209874347164853165236327521, 11.95171543941184700994078855858
