| L(s) = 1 | + 16.2·2-s − 32.8·3-s + 134.·4-s − 220.·5-s − 533.·6-s − 808.·7-s + 111.·8-s − 1.10e3·9-s − 3.58e3·10-s + 6.07e3·11-s − 4.43e3·12-s − 1.21e4·13-s − 1.31e4·14-s + 7.26e3·15-s − 1.54e4·16-s + 1.82e4·17-s − 1.79e4·18-s − 3.51e4·19-s − 2.97e4·20-s + 2.65e4·21-s + 9.85e4·22-s + 1.02e5·23-s − 3.67e3·24-s − 2.93e4·25-s − 1.96e5·26-s + 1.08e5·27-s − 1.09e5·28-s + ⋯ |
| L(s) = 1 | + 1.43·2-s − 0.702·3-s + 1.05·4-s − 0.790·5-s − 1.00·6-s − 0.890·7-s + 0.0771·8-s − 0.505·9-s − 1.13·10-s + 1.37·11-s − 0.740·12-s − 1.52·13-s − 1.27·14-s + 0.555·15-s − 0.943·16-s + 0.902·17-s − 0.724·18-s − 1.17·19-s − 0.832·20-s + 0.626·21-s + 1.97·22-s + 1.76·23-s − 0.0542·24-s − 0.375·25-s − 2.18·26-s + 1.05·27-s − 0.938·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 - 5.06e4T \) |
| good | 2 | \( 1 - 16.2T + 128T^{2} \) |
| 3 | \( 1 + 32.8T + 2.18e3T^{2} \) |
| 5 | \( 1 + 220.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 808.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 6.07e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.21e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.82e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.51e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.02e5T + 3.40e9T^{2} \) |
| 29 | \( 1 + 7.20e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.12e5T + 2.75e10T^{2} \) |
| 41 | \( 1 - 4.30e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.42e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 5.23e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 4.30e4T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.41e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.72e4T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.97e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.20e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.86e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.56e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 3.64e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 6.04e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.03e7T + 8.07e13T^{2} \) |
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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
−14.42474003953665764064339269098, −12.76400463625516371654643484010, −12.12896699990742610644979388322, −11.18214624243433435812405670987, −9.268834589454056107754072882971, −7.04911152480312749664785383468, −5.88671108053809855854367224707, −4.47608395601986158691521920818, −3.12953593043797293650070909570, 0,
3.12953593043797293650070909570, 4.47608395601986158691521920818, 5.88671108053809855854367224707, 7.04911152480312749664785383468, 9.268834589454056107754072882971, 11.18214624243433435812405670987, 12.12896699990742610644979388322, 12.76400463625516371654643484010, 14.42474003953665764064339269098
