| L(s) = 1 | + 2.32·2-s + 6.85·3-s − 2.61·4-s − 20.6·5-s + 15.9·6-s − 15.9·7-s − 24.6·8-s + 20.0·9-s − 47.9·10-s − 0.680·11-s − 17.9·12-s + 45.4·13-s − 37.0·14-s − 141.·15-s − 36.2·16-s − 51.0·17-s + 46.5·18-s − 104.·19-s + 54.0·20-s − 109.·21-s − 1.57·22-s + 41.2·23-s − 168.·24-s + 301.·25-s + 105.·26-s − 47.6·27-s + 41.7·28-s + ⋯ |
| L(s) = 1 | + 0.820·2-s + 1.32·3-s − 0.326·4-s − 1.84·5-s + 1.08·6-s − 0.861·7-s − 1.08·8-s + 0.742·9-s − 1.51·10-s − 0.0186·11-s − 0.431·12-s + 0.968·13-s − 0.706·14-s − 2.43·15-s − 0.566·16-s − 0.728·17-s + 0.609·18-s − 1.25·19-s + 0.604·20-s − 1.13·21-s − 0.0153·22-s + 0.374·23-s − 1.43·24-s + 2.41·25-s + 0.794·26-s − 0.339·27-s + 0.281·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 197 | \( 1 - 197T \) |
| good | 2 | \( 1 - 2.32T + 8T^{2} \) |
| 3 | \( 1 - 6.85T + 27T^{2} \) |
| 5 | \( 1 + 20.6T + 125T^{2} \) |
| 7 | \( 1 + 15.9T + 343T^{2} \) |
| 11 | \( 1 + 0.680T + 1.33e3T^{2} \) |
| 13 | \( 1 - 45.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 51.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 104.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 41.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 168.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 336.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 292.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 186.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 34.6T + 7.95e4T^{2} \) |
| 47 | \( 1 - 189.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 416.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 426.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 752.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 690.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 281.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 525.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 24.0T + 4.93e5T^{2} \) |
| 83 | \( 1 - 50.4T + 5.71e5T^{2} \) |
| 89 | \( 1 - 708.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 786.T + 9.12e5T^{2} \) |
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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
−11.83870095610078472984003202804, −10.72676311288651598733153512628, −9.006859814760997169539851311158, −8.683616216892366521900576888502, −7.60500414113086035863205162704, −6.34327794211513031411646756482, −4.43787876290347514567084479807, −3.72834790174633590375312635623, −2.96311363415539507575319957422, 0,
2.96311363415539507575319957422, 3.72834790174633590375312635623, 4.43787876290347514567084479807, 6.34327794211513031411646756482, 7.60500414113086035863205162704, 8.683616216892366521900576888502, 9.006859814760997169539851311158, 10.72676311288651598733153512628, 11.83870095610078472984003202804
