| L(s) = 1 | + 10.1·2-s + 71.7·4-s + 4.37·5-s + 49·7-s + 404.·8-s + 44.5·10-s + 484.·11-s − 553.·13-s + 499.·14-s + 1.82e3·16-s − 2.14e3·17-s + 2.47e3·19-s + 313.·20-s + 4.93e3·22-s − 1.18e3·23-s − 3.10e3·25-s − 5.63e3·26-s + 3.51e3·28-s − 4.21e3·29-s − 3.49e3·31-s + 5.63e3·32-s − 2.18e4·34-s + 214.·35-s + 1.30e4·37-s + 2.51e4·38-s + 1.77e3·40-s − 8.99e3·41-s + ⋯ |
| L(s) = 1 | + 1.80·2-s + 2.24·4-s + 0.0783·5-s + 0.377·7-s + 2.23·8-s + 0.140·10-s + 1.20·11-s − 0.908·13-s + 0.680·14-s + 1.78·16-s − 1.80·17-s + 1.57·19-s + 0.175·20-s + 2.17·22-s − 0.466·23-s − 0.993·25-s − 1.63·26-s + 0.847·28-s − 0.930·29-s − 0.653·31-s + 0.972·32-s − 3.24·34-s + 0.0295·35-s + 1.57·37-s + 2.82·38-s + 0.174·40-s − 0.835·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.905151526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.905151526\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - 49T \) |
| good | 2 | \( 1 - 10.1T + 32T^{2} \) |
| 5 | \( 1 - 4.37T + 3.12e3T^{2} \) |
| 11 | \( 1 - 484.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 553.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.14e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.47e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.18e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.21e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.49e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.30e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 8.99e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 8.80e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.33e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.57e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.21e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 8.24e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.62e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.66e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.83e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.37e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.17e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.77e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.20e5T + 8.58e9T^{2} \) |
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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
−13.97262707490664337642471793909, −13.05271810904061071637032532023, −11.82449447341273226391472252641, −11.28187423026709113648453042879, −9.463969696362765701217180460823, −7.43233358793865184845821544779, −6.27016798925568475666674988624, −4.94968966841144344398077184180, −3.79932874188218221889854580516, −2.08539342050564086660778295017,
2.08539342050564086660778295017, 3.79932874188218221889854580516, 4.94968966841144344398077184180, 6.27016798925568475666674988624, 7.43233358793865184845821544779, 9.463969696362765701217180460823, 11.28187423026709113648453042879, 11.82449447341273226391472252641, 13.05271810904061071637032532023, 13.97262707490664337642471793909
