| L(s) = 1 | − 61·2-s + 243·3-s + 2.69e3·4-s − 3.12e3·5-s − 1.48e4·6-s − 1.02e5·8-s + 5.90e4·9-s + 1.90e5·10-s + 6.55e5·12-s − 7.59e5·15-s + 3.46e6·16-s + 2.41e6·17-s − 3.60e6·18-s − 2.69e5·19-s − 8.42e6·20-s + 1.09e7·23-s − 2.47e7·24-s + 9.76e6·25-s + 1.43e7·27-s + 4.63e7·30-s + 9.19e6·31-s − 1.06e8·32-s − 1.47e8·34-s + 1.59e8·36-s + 1.64e7·38-s + 3.18e8·40-s − 1.84e8·45-s + ⋯ |
| L(s) = 1 | − 1.90·2-s + 3-s + 2.63·4-s − 5-s − 1.90·6-s − 3.11·8-s + 9-s + 1.90·10-s + 2.63·12-s − 15-s + 3.30·16-s + 1.70·17-s − 1.90·18-s − 0.108·19-s − 2.63·20-s + 1.70·23-s − 3.11·24-s + 25-s + 27-s + 1.90·30-s + 0.321·31-s − 3.18·32-s − 3.24·34-s + 2.63·36-s + 0.207·38-s + 3.11·40-s − 45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.9153420569\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9153420569\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p^{5} T \) |
| 5 | \( 1 + p^{5} T \) |
| good | 2 | \( 1 + 61 T + p^{10} T^{2} \) |
| 7 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 11 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 13 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 17 | \( 1 - 2419214 T + p^{10} T^{2} \) |
| 19 | \( 1 + 269302 T + p^{10} T^{2} \) |
| 23 | \( 1 - 10950686 T + p^{10} T^{2} \) |
| 29 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 31 | \( 1 - 9196802 T + p^{10} T^{2} \) |
| 37 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 41 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 43 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 47 | \( 1 - 311808014 T + p^{10} T^{2} \) |
| 53 | \( 1 + 836229514 T + p^{10} T^{2} \) |
| 59 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 61 | \( 1 + 478013398 T + p^{10} T^{2} \) |
| 67 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 71 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 73 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 79 | \( 1 + 1245148702 T + p^{10} T^{2} \) |
| 83 | \( 1 - 2642233286 T + p^{10} T^{2} \) |
| 89 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 97 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
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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
−16.96756951583670994642867557593, −15.81850681480840723747168013083, −14.80903222775242165827424205910, −12.24534599630655849095306644176, −10.68998041863222359155659405731, −9.297624764221187081177438129819, −8.125345839472678119082833585447, −7.19928522840764006528825645446, −3.05824866706885226230988980804, −1.04101687092716463084980436862,
1.04101687092716463084980436862, 3.05824866706885226230988980804, 7.19928522840764006528825645446, 8.125345839472678119082833585447, 9.297624764221187081177438129819, 10.68998041863222359155659405731, 12.24534599630655849095306644176, 14.80903222775242165827424205910, 15.81850681480840723747168013083, 16.96756951583670994642867557593
