| L(s) = 1 | + 5.09·2-s − 7.11·3-s + 17.9·4-s + 8.87·5-s − 36.2·6-s − 1.96·7-s + 50.5·8-s + 23.6·9-s + 45.1·10-s + 42.4·11-s − 127.·12-s + 41.5·13-s − 10.0·14-s − 63.1·15-s + 113.·16-s + 12.7·17-s + 120.·18-s + 52.0·19-s + 159.·20-s + 14.0·21-s + 215.·22-s + 43.8·23-s − 359.·24-s − 46.1·25-s + 211.·26-s + 23.9·27-s − 35.2·28-s + ⋯ |
| L(s) = 1 | + 1.79·2-s − 1.36·3-s + 2.23·4-s + 0.794·5-s − 2.46·6-s − 0.106·7-s + 2.23·8-s + 0.875·9-s + 1.42·10-s + 1.16·11-s − 3.06·12-s + 0.886·13-s − 0.191·14-s − 1.08·15-s + 1.77·16-s + 0.181·17-s + 1.57·18-s + 0.629·19-s + 1.77·20-s + 0.145·21-s + 2.09·22-s + 0.397·23-s − 3.05·24-s − 0.369·25-s + 1.59·26-s + 0.170·27-s − 0.238·28-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
| \(L(2)\) |
\(\approx\) |
\(3.923114489\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.923114489\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 5.09T + 8T^{2} \) |
| 3 | \( 1 + 7.11T + 27T^{2} \) |
| 5 | \( 1 - 8.87T + 125T^{2} \) |
| 7 | \( 1 + 1.96T + 343T^{2} \) |
| 11 | \( 1 - 42.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 41.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 12.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 52.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 43.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 233.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 15.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 215.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 240.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 72.1T + 7.95e4T^{2} \) |
| 47 | \( 1 - 249.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 656.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 282.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 705.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 601.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 839.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 194.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 450.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 156.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 315.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.20e3T + 9.12e5T^{2} \) |
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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
−12.10316267388592749738123433328, −11.40009914109244724409352672778, −10.70797245811444177713983360544, −9.329456048117356419067369345783, −7.18547363262216600523407791236, −6.10318898452810479430865726286, −5.83033384430894628224667883889, −4.68604179011077023694566456704, −3.46347946582898727283669312996, −1.54993107992247367298299613348,
1.54993107992247367298299613348, 3.46347946582898727283669312996, 4.68604179011077023694566456704, 5.83033384430894628224667883889, 6.10318898452810479430865726286, 7.18547363262216600523407791236, 9.329456048117356419067369345783, 10.70797245811444177713983360544, 11.40009914109244724409352672778, 12.10316267388592749738123433328
