| L(s) = 1 | − 4·2-s − 2.13·3-s + 16·4-s − 10.7·5-s + 8.54·6-s − 195.·7-s − 64·8-s − 238.·9-s + 43.1·10-s − 17.0·11-s − 34.1·12-s + 263.·13-s + 780.·14-s + 23.0·15-s + 256·16-s − 577.·17-s + 953.·18-s − 172.·20-s + 416.·21-s + 68.3·22-s − 565.·23-s + 136.·24-s − 3.00e3·25-s − 1.05e3·26-s + 1.02e3·27-s − 3.12e3·28-s − 6.71e3·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.137·3-s + 0.5·4-s − 0.193·5-s + 0.0969·6-s − 1.50·7-s − 0.353·8-s − 0.981·9-s + 0.136·10-s − 0.0425·11-s − 0.0685·12-s + 0.431·13-s + 1.06·14-s + 0.0264·15-s + 0.250·16-s − 0.484·17-s + 0.693·18-s − 0.0965·20-s + 0.206·21-s + 0.0301·22-s − 0.222·23-s + 0.0484·24-s − 0.962·25-s − 0.305·26-s + 0.271·27-s − 0.752·28-s − 1.48·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.05077040371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05077040371\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2.13T + 243T^{2} \) |
| 5 | \( 1 + 10.7T + 3.12e3T^{2} \) |
| 7 | \( 1 + 195.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 17.0T + 1.61e5T^{2} \) |
| 13 | \( 1 - 263.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 577.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 565.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.71e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.28e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.71e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.21e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.54e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.49e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.56e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.97e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.14e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.19e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.74e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.56e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.34e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.68e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.60e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.30e5T + 8.58e9T^{2} \) |
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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
−9.459296245757175557226805349701, −8.999249967760977570580278099089, −7.995108289602508878188436337708, −7.04210746368692396225120384201, −6.19321200131412298228283771018, −5.51496204997283007199299264928, −3.80659093148985787987984249576, −3.06422179737781007131984212267, −1.83523915388917531764875699961, −0.11081705564925852662455982043,
0.11081705564925852662455982043, 1.83523915388917531764875699961, 3.06422179737781007131984212267, 3.80659093148985787987984249576, 5.51496204997283007199299264928, 6.19321200131412298228283771018, 7.04210746368692396225120384201, 7.995108289602508878188436337708, 8.999249967760977570580278099089, 9.459296245757175557226805349701
