| L(s) = 1 | − 19.0·3-s − 9.45·5-s − 38.2·7-s + 120.·9-s + 7.22·11-s + 889.·13-s + 180.·15-s + 1.47e3·17-s + 276.·19-s + 728.·21-s + 984.·23-s − 3.03e3·25-s + 2.34e3·27-s + 769.·29-s − 668.·31-s − 137.·33-s + 361.·35-s + 6.27e3·37-s − 1.69e4·39-s + 1.39e4·41-s − 1.45e4·43-s − 1.13e3·45-s + 1.83e4·47-s − 1.53e4·49-s − 2.80e4·51-s − 1.96e4·53-s − 68.3·55-s + ⋯ |
| L(s) = 1 | − 1.22·3-s − 0.169·5-s − 0.294·7-s + 0.494·9-s + 0.0180·11-s + 1.46·13-s + 0.206·15-s + 1.23·17-s + 0.175·19-s + 0.360·21-s + 0.387·23-s − 0.971·25-s + 0.617·27-s + 0.169·29-s − 0.125·31-s − 0.0220·33-s + 0.0498·35-s + 0.754·37-s − 1.78·39-s + 1.29·41-s − 1.20·43-s − 0.0836·45-s + 1.21·47-s − 0.912·49-s − 1.50·51-s − 0.962·53-s − 0.00304·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.286513238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.286513238\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 257 | \( 1 + 6.60e4T \) |
| good | 3 | \( 1 + 19.0T + 243T^{2} \) |
| 5 | \( 1 + 9.45T + 3.12e3T^{2} \) |
| 7 | \( 1 + 38.2T + 1.68e4T^{2} \) |
| 11 | \( 1 - 7.22T + 1.61e5T^{2} \) |
| 13 | \( 1 - 889.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.47e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 276.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 984.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 769.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 668.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.27e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.39e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.45e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.83e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.96e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.19e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 9.34e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 7.93e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.18e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.54e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.34e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.48e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.57e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.49e3T + 8.58e9T^{2} \) |
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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
−9.337531062549667498428231543683, −8.300331589364237312548962561293, −7.48556736981017021781597447162, −6.33747752414729768331244051158, −5.93868816126441945674167875589, −5.07593323384471963513571537758, −3.97537017535996553530750221037, −3.05496718968963039237079810693, −1.41746208818756220670675141666, −0.57503368713089878001970119154,
0.57503368713089878001970119154, 1.41746208818756220670675141666, 3.05496718968963039237079810693, 3.97537017535996553530750221037, 5.07593323384471963513571537758, 5.93868816126441945674167875589, 6.33747752414729768331244051158, 7.48556736981017021781597447162, 8.300331589364237312548962561293, 9.337531062549667498428231543683
