| L(s) = 1 | − 16·2-s + 256·4-s − 1.16e3·5-s + 6.24e3·7-s − 4.09e3·8-s + 1.87e4·10-s − 1.65e4·11-s − 1.36e4·13-s − 9.99e4·14-s + 6.55e4·16-s − 1.56e5·17-s + 5.43e5·19-s − 2.99e5·20-s + 2.64e5·22-s − 1.46e6·23-s − 5.85e5·25-s + 2.17e5·26-s + 1.59e6·28-s + 4.58e6·29-s + 6.53e6·31-s − 1.04e6·32-s + 2.51e6·34-s − 7.30e6·35-s − 1.36e7·37-s − 8.69e6·38-s + 4.78e6·40-s + 1.54e7·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.836·5-s + 0.983·7-s − 0.353·8-s + 0.591·10-s − 0.340·11-s − 0.132·13-s − 0.695·14-s + 0.250·16-s − 0.455·17-s + 0.957·19-s − 0.418·20-s + 0.240·22-s − 1.09·23-s − 0.300·25-s + 0.0935·26-s + 0.491·28-s + 1.20·29-s + 1.27·31-s − 0.176·32-s + 0.322·34-s − 0.822·35-s − 1.19·37-s − 0.676·38-s + 0.295·40-s + 0.853·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 16T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 1.16e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 6.24e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 1.65e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.36e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 1.56e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 5.43e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.46e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.58e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 6.53e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.36e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.54e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.84e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 6.05e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 4.91e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.08e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.38e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.06e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.07e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 8.53e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.52e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 1.64e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 3.89e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.39e7T + 7.60e17T^{2} \) |
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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
−10.68217765472357151559628221228, −9.661529076241530694531667440767, −8.289198378616620493651670626622, −7.913750214644632366249903933669, −6.73257471921995611735010480673, −5.24170743390486726607298034989, −4.04550229547493976520762154298, −2.56247946271949633876211853578, −1.23011790730964835828015019922, 0,
1.23011790730964835828015019922, 2.56247946271949633876211853578, 4.04550229547493976520762154298, 5.24170743390486726607298034989, 6.73257471921995611735010480673, 7.913750214644632366249903933669, 8.289198378616620493651670626622, 9.661529076241530694531667440767, 10.68217765472357151559628221228
