| L(s) = 1 | − 24·2-s + 81·3-s + 64·4-s − 144·5-s − 1.94e3·6-s + 2.40e3·7-s + 1.07e4·8-s + 6.56e3·9-s + 3.45e3·10-s − 1.50e4·11-s + 5.18e3·12-s − 1.51e5·13-s − 5.76e4·14-s − 1.16e4·15-s − 2.90e5·16-s − 3.50e5·17-s − 1.57e5·18-s − 6.91e5·19-s − 9.21e3·20-s + 1.94e5·21-s + 3.60e5·22-s + 8.92e5·23-s + 8.70e5·24-s − 1.93e6·25-s + 3.63e6·26-s + 5.31e5·27-s + 1.53e5·28-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 0.577·3-s + 1/8·4-s − 0.103·5-s − 0.612·6-s + 0.377·7-s + 0.928·8-s + 1/3·9-s + 0.109·10-s − 0.309·11-s + 0.0721·12-s − 1.47·13-s − 0.400·14-s − 0.0594·15-s − 1.10·16-s − 1.01·17-s − 0.353·18-s − 1.21·19-s − 0.0128·20-s + 0.218·21-s + 0.328·22-s + 0.664·23-s + 0.535·24-s − 0.989·25-s + 1.56·26-s + 0.192·27-s + 0.0472·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p^{4} T \) |
| 7 | \( 1 - p^{4} T \) |
| good | 2 | \( 1 + 3 p^{3} T + p^{9} T^{2} \) |
| 5 | \( 1 + 144 T + p^{9} T^{2} \) |
| 11 | \( 1 + 15030 T + p^{9} T^{2} \) |
| 13 | \( 1 + 151486 T + p^{9} T^{2} \) |
| 17 | \( 1 + 350448 T + p^{9} T^{2} \) |
| 19 | \( 1 + 691108 T + p^{9} T^{2} \) |
| 23 | \( 1 - 892458 T + p^{9} T^{2} \) |
| 29 | \( 1 - 1648518 T + p^{9} T^{2} \) |
| 31 | \( 1 + 3734296 T + p^{9} T^{2} \) |
| 37 | \( 1 + 11471902 T + p^{9} T^{2} \) |
| 41 | \( 1 - 13985724 T + p^{9} T^{2} \) |
| 43 | \( 1 - 16794524 T + p^{9} T^{2} \) |
| 47 | \( 1 + 14012052 T + p^{9} T^{2} \) |
| 53 | \( 1 + 97439910 T + p^{9} T^{2} \) |
| 59 | \( 1 - 110798304 T + p^{9} T^{2} \) |
| 61 | \( 1 + 93816682 T + p^{9} T^{2} \) |
| 67 | \( 1 + 122446456 T + p^{9} T^{2} \) |
| 71 | \( 1 - 206197398 T + p^{9} T^{2} \) |
| 73 | \( 1 - 250337558 T + p^{9} T^{2} \) |
| 79 | \( 1 + 38314852 T + p^{9} T^{2} \) |
| 83 | \( 1 + 514086924 T + p^{9} T^{2} \) |
| 89 | \( 1 + 1061294916 T + p^{9} T^{2} \) |
| 97 | \( 1 + 73841578 T + p^{9} T^{2} \) |
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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
−15.48607158978069518914120794096, −14.20618675043846378235634351190, −12.79198752602841770933342871347, −10.89600158903245920928066823991, −9.605955238251057207401876588462, −8.448328542488532790545873205311, −7.26239420724494437041848791891, −4.57686365118691623409371195264, −2.09925926393890646695166835056, 0,
2.09925926393890646695166835056, 4.57686365118691623409371195264, 7.26239420724494437041848791891, 8.448328542488532790545873205311, 9.605955238251057207401876588462, 10.89600158903245920928066823991, 12.79198752602841770933342871347, 14.20618675043846378235634351190, 15.48607158978069518914120794096
