| L(s) = 1 | − 1.59·2-s − 8.64·3-s − 5.45·4-s + 11.1·5-s + 13.7·6-s + 7·7-s + 21.4·8-s + 47.7·9-s − 17.7·10-s − 11.0·11-s + 47.1·12-s + 7.81·13-s − 11.1·14-s − 95.9·15-s + 9.42·16-s − 45.3·17-s − 76.1·18-s − 5.00·19-s − 60.5·20-s − 60.5·21-s + 17.6·22-s − 23·23-s − 185.·24-s − 1.76·25-s − 12.4·26-s − 179.·27-s − 38.1·28-s + ⋯ |
| L(s) = 1 | − 0.563·2-s − 1.66·3-s − 0.682·4-s + 0.992·5-s + 0.938·6-s + 0.377·7-s + 0.948·8-s + 1.76·9-s − 0.559·10-s − 0.303·11-s + 1.13·12-s + 0.166·13-s − 0.213·14-s − 1.65·15-s + 0.147·16-s − 0.647·17-s − 0.996·18-s − 0.0604·19-s − 0.677·20-s − 0.628·21-s + 0.170·22-s − 0.208·23-s − 1.57·24-s − 0.0140·25-s − 0.0940·26-s − 1.27·27-s − 0.257·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 7T \) |
| 23 | \( 1 + 23T \) |
| good | 2 | \( 1 + 1.59T + 8T^{2} \) |
| 3 | \( 1 + 8.64T + 27T^{2} \) |
| 5 | \( 1 - 11.1T + 125T^{2} \) |
| 11 | \( 1 + 11.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 7.81T + 2.19e3T^{2} \) |
| 17 | \( 1 + 45.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 5.00T + 6.85e3T^{2} \) |
| 29 | \( 1 + 164.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 19.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 247.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 211.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 417.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 431.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 366.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 595.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 406.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 478.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 879.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 66.7T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.21e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.35e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 228.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 768.T + 9.12e5T^{2} \) |
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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
−11.68210056327342000559263061110, −10.77024700501191045966067951121, −10.03927660017257360196391608718, −9.096429534082530348866448519367, −7.66728553438251851464320241972, −6.28930905252497771126092456970, −5.40483863448746523918415306943, −4.45300354138034084273989391449, −1.55620877715818817571681823830, 0,
1.55620877715818817571681823830, 4.45300354138034084273989391449, 5.40483863448746523918415306943, 6.28930905252497771126092456970, 7.66728553438251851464320241972, 9.096429534082530348866448519367, 10.03927660017257360196391608718, 10.77024700501191045966067951121, 11.68210056327342000559263061110
