| L(s) = 1 | − 282.·2-s − 6.56e3·3-s − 5.15e4·4-s − 1.29e6·5-s + 1.85e6·6-s − 5.76e6·7-s + 5.14e7·8-s + 4.30e7·9-s + 3.66e8·10-s + 2.68e8·11-s + 3.38e8·12-s + 3.66e8·13-s + 1.62e9·14-s + 8.51e9·15-s − 7.77e9·16-s − 7.60e9·17-s − 1.21e10·18-s + 5.73e10·19-s + 6.68e10·20-s + 3.78e10·21-s − 7.56e10·22-s + 5.11e11·23-s − 3.37e11·24-s + 9.22e11·25-s − 1.03e11·26-s − 2.82e11·27-s + 2.97e11·28-s + ⋯ |
| L(s) = 1 | − 0.779·2-s − 0.577·3-s − 0.393·4-s − 1.48·5-s + 0.449·6-s − 0.377·7-s + 1.08·8-s + 0.333·9-s + 1.15·10-s + 0.377·11-s + 0.226·12-s + 0.124·13-s + 0.294·14-s + 0.858·15-s − 0.452·16-s − 0.264·17-s − 0.259·18-s + 0.775·19-s + 0.584·20-s + 0.218·21-s − 0.294·22-s + 1.36·23-s − 0.626·24-s + 1.20·25-s − 0.0971·26-s − 0.192·27-s + 0.148·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 6.56e3T \) |
| 7 | \( 1 + 5.76e6T \) |
| good | 2 | \( 1 + 282.T + 1.31e5T^{2} \) |
| 5 | \( 1 + 1.29e6T + 7.62e11T^{2} \) |
| 11 | \( 1 - 2.68e8T + 5.05e17T^{2} \) |
| 13 | \( 1 - 3.66e8T + 8.65e18T^{2} \) |
| 17 | \( 1 + 7.60e9T + 8.27e20T^{2} \) |
| 19 | \( 1 - 5.73e10T + 5.48e21T^{2} \) |
| 23 | \( 1 - 5.11e11T + 1.41e23T^{2} \) |
| 29 | \( 1 + 2.84e12T + 7.25e24T^{2} \) |
| 31 | \( 1 - 5.40e12T + 2.25e25T^{2} \) |
| 37 | \( 1 - 1.26e13T + 4.56e26T^{2} \) |
| 41 | \( 1 + 5.19e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 1.28e14T + 5.87e27T^{2} \) |
| 47 | \( 1 - 2.02e14T + 2.66e28T^{2} \) |
| 53 | \( 1 + 6.66e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 5.74e14T + 1.27e30T^{2} \) |
| 61 | \( 1 - 5.34e14T + 2.24e30T^{2} \) |
| 67 | \( 1 + 9.71e14T + 1.10e31T^{2} \) |
| 71 | \( 1 + 5.11e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 1.23e16T + 4.74e31T^{2} \) |
| 79 | \( 1 - 1.48e16T + 1.81e32T^{2} \) |
| 83 | \( 1 - 2.00e16T + 4.21e32T^{2} \) |
| 89 | \( 1 - 1.93e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 5.62e16T + 5.95e33T^{2} \) |
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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
−13.33461342726013233954879546034, −11.93662827330318849475939424534, −10.88676891104451416124967831146, −9.415986201442231249603234916865, −8.112586412605847298387746225309, −6.95908474465170954115912332364, −4.86071467367298850383028452415, −3.60222866695541655438669479605, −1.01083736104521162868872424268, 0,
1.01083736104521162868872424268, 3.60222866695541655438669479605, 4.86071467367298850383028452415, 6.95908474465170954115912332364, 8.112586412605847298387746225309, 9.415986201442231249603234916865, 10.88676891104451416124967831146, 11.93662827330318849475939424534, 13.33461342726013233954879546034
