| L(s) = 1 | − 16·2-s − 81·3-s + 256·4-s − 866·5-s + 1.29e3·6-s − 4.79e3·7-s − 4.09e3·8-s + 6.56e3·9-s + 1.38e4·10-s + 9.11e3·11-s − 2.07e4·12-s + 7.67e4·14-s + 7.01e4·15-s + 6.55e4·16-s − 1.04e5·18-s − 2.21e5·20-s + 3.88e5·21-s − 1.45e5·22-s + 3.31e5·24-s + 3.59e5·25-s − 5.31e5·27-s − 1.22e6·28-s + 7.45e5·29-s − 1.12e6·30-s − 1.61e6·31-s − 1.04e6·32-s − 7.38e5·33-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 4-s − 1.38·5-s + 6-s − 1.99·7-s − 8-s + 9-s + 1.38·10-s + 0.622·11-s − 12-s + 1.99·14-s + 1.38·15-s + 16-s − 18-s − 1.38·20-s + 1.99·21-s − 0.622·22-s + 24-s + 0.919·25-s − 27-s − 1.99·28-s + 1.05·29-s − 1.38·30-s − 1.75·31-s − 32-s − 0.622·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.2679836411\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2679836411\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{4} T \) |
| 3 | \( 1 + p^{4} T \) |
| good | 5 | \( 1 + 866 T + p^{8} T^{2} \) |
| 7 | \( 1 + 4798 T + p^{8} T^{2} \) |
| 11 | \( 1 - 9118 T + p^{8} T^{2} \) |
| 13 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 17 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 19 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 23 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 29 | \( 1 - 745438 T + p^{8} T^{2} \) |
| 31 | \( 1 + 1618558 T + p^{8} T^{2} \) |
| 37 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 41 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 43 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 47 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 53 | \( 1 - 5425438 T + p^{8} T^{2} \) |
| 59 | \( 1 + 22852322 T + p^{8} T^{2} \) |
| 61 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 67 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 71 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 73 | \( 1 - 9756482 T + p^{8} T^{2} \) |
| 79 | \( 1 - 5237762 T + p^{8} T^{2} \) |
| 83 | \( 1 - 77460958 T + p^{8} T^{2} \) |
| 89 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 97 | \( 1 - 121608962 T + p^{8} T^{2} \) |
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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
−16.15621479590326232233898062940, −15.40700836189711710309712745834, −12.64742323669699213695779219941, −11.83157488138598237393357434009, −10.53649148619754362258249523839, −9.243242250556266741226433073685, −7.32113473444828856020806993270, −6.30695919810997142288053028637, −3.58163988038719878136511937771, −0.48671599862018099230745548647,
0.48671599862018099230745548647, 3.58163988038719878136511937771, 6.30695919810997142288053028637, 7.32113473444828856020806993270, 9.243242250556266741226433073685, 10.53649148619754362258249523839, 11.83157488138598237393357434009, 12.64742323669699213695779219941, 15.40700836189711710309712745834, 16.15621479590326232233898062940
