| L(s) = 1 | − 4.30·3-s + 28.3·7-s − 8.49·9-s + 65.2·11-s + 33.6·13-s + 73.3·17-s + 134.·19-s − 121.·21-s − 14.7·23-s + 152.·27-s + 224.·29-s − 68.8·31-s − 280.·33-s + 196.·37-s − 144.·39-s − 143.·41-s − 15.0·43-s + 134.·47-s + 458.·49-s − 315.·51-s − 262.·53-s − 576.·57-s − 119.·59-s − 16.5·61-s − 240.·63-s − 545.·67-s + 63.2·69-s + ⋯ |
| L(s) = 1 | − 0.827·3-s + 1.52·7-s − 0.314·9-s + 1.78·11-s + 0.718·13-s + 1.04·17-s + 1.61·19-s − 1.26·21-s − 0.133·23-s + 1.08·27-s + 1.43·29-s − 0.398·31-s − 1.48·33-s + 0.872·37-s − 0.594·39-s − 0.545·41-s − 0.0534·43-s + 0.417·47-s + 1.33·49-s − 0.865·51-s − 0.681·53-s − 1.34·57-s − 0.264·59-s − 0.0347·61-s − 0.481·63-s − 0.994·67-s + 0.110·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.688151991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.688151991\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 4.30T + 27T^{2} \) |
| 7 | \( 1 - 28.3T + 343T^{2} \) |
| 11 | \( 1 - 65.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 33.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 73.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 134.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 14.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 224.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 68.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 196.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 143.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 15.0T + 7.95e4T^{2} \) |
| 47 | \( 1 - 134.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 262.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 119.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 16.5T + 2.26e5T^{2} \) |
| 67 | \( 1 + 545.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 199.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 43.2T + 3.89e5T^{2} \) |
| 79 | \( 1 + 438.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.22e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 723.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.13e3T + 9.12e5T^{2} \) |
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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
−8.963910164958330182926780258343, −8.267621130417366345140788621419, −7.45830538157350921783095114534, −6.47634408627750754927807395235, −5.73779205113317063384227396582, −5.01898323509557354295304327767, −4.15514285484358573213765570967, −3.09342796937854449034142240949, −1.41795931004164395333989723752, −1.00255278021045537620902070408,
1.00255278021045537620902070408, 1.41795931004164395333989723752, 3.09342796937854449034142240949, 4.15514285484358573213765570967, 5.01898323509557354295304327767, 5.73779205113317063384227396582, 6.47634408627750754927807395235, 7.45830538157350921783095114534, 8.267621130417366345140788621419, 8.963910164958330182926780258343
