| L(s) = 1 | + 4.61·2-s − 3·3-s + 13.3·4-s − 3.21·5-s − 13.8·6-s + 15.9·7-s + 24.5·8-s + 9·9-s − 14.8·10-s + 54.2·11-s − 39.9·12-s + 85.2·13-s + 73.8·14-s + 9.65·15-s + 6.85·16-s − 48.1·17-s + 41.5·18-s + 64.2·19-s − 42.8·20-s − 47.9·21-s + 250.·22-s − 191.·23-s − 73.6·24-s − 114.·25-s + 393.·26-s − 27·27-s + 212.·28-s + ⋯ |
| L(s) = 1 | + 1.63·2-s − 0.577·3-s + 1.66·4-s − 0.287·5-s − 0.942·6-s + 0.863·7-s + 1.08·8-s + 0.333·9-s − 0.469·10-s + 1.48·11-s − 0.961·12-s + 1.81·13-s + 1.40·14-s + 0.166·15-s + 0.107·16-s − 0.686·17-s + 0.544·18-s + 0.775·19-s − 0.479·20-s − 0.498·21-s + 2.42·22-s − 1.73·23-s − 0.626·24-s − 0.917·25-s + 2.97·26-s − 0.192·27-s + 1.43·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.893780610\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.893780610\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 59 | \( 1 + 59T \) |
| good | 2 | \( 1 - 4.61T + 8T^{2} \) |
| 5 | \( 1 + 3.21T + 125T^{2} \) |
| 7 | \( 1 - 15.9T + 343T^{2} \) |
| 11 | \( 1 - 54.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 85.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 48.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 64.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 191.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 15.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 209.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 418.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 226.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 207.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 330.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 449.T + 1.48e5T^{2} \) |
| 61 | \( 1 + 393.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 67.7T + 3.00e5T^{2} \) |
| 71 | \( 1 + 589.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 229.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 563.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.17e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.33e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.36e3T + 9.12e5T^{2} \) |
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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
−12.14784755588953020622399273541, −11.47413776618721293590312900219, −11.01997975403699035975123182291, −9.221957578574429908519148552599, −7.79489349199683578610563066916, −6.37496078470151327749017481618, −5.80362385700175039609418471032, −4.34699508255869451716745833988, −3.76182584547269052236889615909, −1.62018724410396158492005331826,
1.62018724410396158492005331826, 3.76182584547269052236889615909, 4.34699508255869451716745833988, 5.80362385700175039609418471032, 6.37496078470151327749017481618, 7.79489349199683578610563066916, 9.221957578574429908519148552599, 11.01997975403699035975123182291, 11.47413776618721293590312900219, 12.14784755588953020622399273541
