| L(s) = 1 | − 7.82·3-s − 9.23·5-s + 34.1·9-s + 71.3·11-s − 46.3·13-s + 72.2·15-s − 83.7·17-s − 2.16·19-s − 193.·23-s − 39.6·25-s − 56.2·27-s − 135.·29-s + 20.5·31-s − 557.·33-s − 324.·37-s + 362.·39-s + 431.·41-s + 135.·43-s − 315.·45-s − 592.·47-s + 655.·51-s + 182.·53-s − 658.·55-s + 16.9·57-s + 208.·59-s + 80.2·61-s + 428.·65-s + ⋯ |
| L(s) = 1 | − 1.50·3-s − 0.826·5-s + 1.26·9-s + 1.95·11-s − 0.989·13-s + 1.24·15-s − 1.19·17-s − 0.0261·19-s − 1.75·23-s − 0.317·25-s − 0.400·27-s − 0.868·29-s + 0.119·31-s − 2.94·33-s − 1.44·37-s + 1.48·39-s + 1.64·41-s + 0.481·43-s − 1.04·45-s − 1.83·47-s + 1.79·51-s + 0.473·53-s − 1.61·55-s + 0.0393·57-s + 0.460·59-s + 0.168·61-s + 0.817·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5253121332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5253121332\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 7.82T + 27T^{2} \) |
| 5 | \( 1 + 9.23T + 125T^{2} \) |
| 11 | \( 1 - 71.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 46.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 83.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 2.16T + 6.85e3T^{2} \) |
| 23 | \( 1 + 193.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 135.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 20.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 324.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 431.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 135.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 592.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 182.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 208.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 80.2T + 2.26e5T^{2} \) |
| 67 | \( 1 - 831.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 59.1T + 3.57e5T^{2} \) |
| 73 | \( 1 - 367.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 438.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 35.0T + 5.71e5T^{2} \) |
| 89 | \( 1 + 824.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.29e3T + 9.12e5T^{2} \) |
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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
−9.979094906896382514616343580758, −9.222374197566996186937199572728, −8.075028215082997481981381824360, −6.98621961340507810244804032841, −6.47453753227176582216253106967, −5.52523206065058821021720892141, −4.38807382532300646349041386172, −3.87382796753937422501062035415, −1.88112349186905605793690316091, −0.43372580611819711426856270273,
0.43372580611819711426856270273, 1.88112349186905605793690316091, 3.87382796753937422501062035415, 4.38807382532300646349041386172, 5.52523206065058821021720892141, 6.47453753227176582216253106967, 6.98621961340507810244804032841, 8.075028215082997481981381824360, 9.222374197566996186937199572728, 9.979094906896382514616343580758
