| L(s) = 1 | − 0.790·2-s − 9·3-s − 31.3·4-s + 104.·5-s + 7.11·6-s + 50.1·8-s + 81·9-s − 82.3·10-s − 497.·11-s + 282.·12-s + 206.·13-s − 937.·15-s + 964.·16-s − 63.1·17-s − 64.0·18-s − 1.32e3·19-s − 3.26e3·20-s + 393.·22-s − 194.·23-s − 451.·24-s + 7.73e3·25-s − 163.·26-s − 729·27-s + 4.32e3·29-s + 741.·30-s + 7.52e3·31-s − 2.36e3·32-s + ⋯ |
| L(s) = 1 | − 0.139·2-s − 0.577·3-s − 0.980·4-s + 1.86·5-s + 0.0807·6-s + 0.276·8-s + 0.333·9-s − 0.260·10-s − 1.24·11-s + 0.566·12-s + 0.338·13-s − 1.07·15-s + 0.941·16-s − 0.0530·17-s − 0.0465·18-s − 0.841·19-s − 1.82·20-s + 0.173·22-s − 0.0766·23-s − 0.159·24-s + 2.47·25-s − 0.0473·26-s − 0.192·27-s + 0.954·29-s + 0.150·30-s + 1.40·31-s − 0.408·32-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
| \(L(3)\) |
\(\approx\) |
\(1.526964007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.526964007\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 0.790T + 32T^{2} \) |
| 5 | \( 1 - 104.T + 3.12e3T^{2} \) |
| 11 | \( 1 + 497.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 206.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 63.1T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.32e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 194.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.32e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.52e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.03e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 4.18e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.96e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 4.38e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.77e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.50e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.06e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.32e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.88e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.13e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.94e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.02e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.12e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.03e4T + 8.58e9T^{2} \) |
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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
−12.46155027305803217576526036857, −10.67761892417320606847678801582, −10.15127395170766728494285441830, −9.248020099471478496805133942293, −8.136909881077012312757723096259, −6.40089641971576003044274838613, −5.54665988761084605530048122176, −4.61462153694137126622245428274, −2.47598522849323675211513679697, −0.888534743569429170056968536288,
0.888534743569429170056968536288, 2.47598522849323675211513679697, 4.61462153694137126622245428274, 5.54665988761084605530048122176, 6.40089641971576003044274838613, 8.136909881077012312757723096259, 9.248020099471478496805133942293, 10.15127395170766728494285441830, 10.67761892417320606847678801582, 12.46155027305803217576526036857
