| L(s) = 1 | + 2.74·2-s − 9·3-s − 24.4·4-s − 58.3·5-s − 24.7·6-s − 155.·8-s + 81·9-s − 160.·10-s + 17.4·11-s + 220.·12-s − 889.·13-s + 525.·15-s + 356.·16-s − 1.02e3·17-s + 222.·18-s + 1.73e3·19-s + 1.42e3·20-s + 47.8·22-s + 3.93e3·23-s + 1.39e3·24-s + 281.·25-s − 2.44e3·26-s − 729·27-s + 5.63e3·29-s + 1.44e3·30-s + 3.09e3·31-s + 5.94e3·32-s + ⋯ |
| L(s) = 1 | + 0.485·2-s − 0.577·3-s − 0.763·4-s − 1.04·5-s − 0.280·6-s − 0.856·8-s + 0.333·9-s − 0.507·10-s + 0.0434·11-s + 0.441·12-s − 1.46·13-s + 0.602·15-s + 0.347·16-s − 0.861·17-s + 0.161·18-s + 1.10·19-s + 0.797·20-s + 0.0210·22-s + 1.55·23-s + 0.494·24-s + 0.0901·25-s − 0.709·26-s − 0.192·27-s + 1.24·29-s + 0.292·30-s + 0.578·31-s + 1.02·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8468921586\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8468921586\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.74T + 32T^{2} \) |
| 5 | \( 1 + 58.3T + 3.12e3T^{2} \) |
| 11 | \( 1 - 17.4T + 1.61e5T^{2} \) |
| 13 | \( 1 + 889.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.02e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.73e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.93e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.63e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.09e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.02e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.83e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.63e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 9.60e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.32e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.60e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.28e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.70e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.59e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.93e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.84e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.58e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.65e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.59e4T + 8.58e9T^{2} \) |
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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
−12.10780483935309903699507647323, −11.50796622696280109871699436607, −10.13077000959666519128151892514, −9.067724889773129116471077827421, −7.83156994860384442958808611183, −6.72559631578348948887831813645, −5.12261466717173574189736814232, −4.49842324787841682048731061753, −3.09704172136941538112104547478, −0.57202263421052946958012512823,
0.57202263421052946958012512823, 3.09704172136941538112104547478, 4.49842324787841682048731061753, 5.12261466717173574189736814232, 6.72559631578348948887831813645, 7.83156994860384442958808611183, 9.067724889773129116471077827421, 10.13077000959666519128151892514, 11.50796622696280109871699436607, 12.10780483935309903699507647323
