| L(s) = 1 | + 10.6·2-s − 19.2·3-s + 80.6·4-s − 25·5-s − 204.·6-s + 113.·7-s + 516.·8-s + 127.·9-s − 265.·10-s + 121·11-s − 1.55e3·12-s − 386.·13-s + 1.20e3·14-s + 481.·15-s + 2.90e3·16-s − 1.37e3·17-s + 1.35e3·18-s + 361·19-s − 2.01e3·20-s − 2.19e3·21-s + 1.28e3·22-s − 2.84e3·23-s − 9.95e3·24-s + 625·25-s − 4.10e3·26-s + 2.21e3·27-s + 9.18e3·28-s + ⋯ |
| L(s) = 1 | + 1.87·2-s − 1.23·3-s + 2.52·4-s − 0.447·5-s − 2.31·6-s + 0.877·7-s + 2.85·8-s + 0.526·9-s − 0.839·10-s + 0.301·11-s − 3.11·12-s − 0.634·13-s + 1.64·14-s + 0.552·15-s + 2.83·16-s − 1.15·17-s + 0.987·18-s + 0.229·19-s − 1.12·20-s − 1.08·21-s + 0.565·22-s − 1.12·23-s − 3.52·24-s + 0.200·25-s − 1.19·26-s + 0.585·27-s + 2.21·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 - 10.6T + 32T^{2} \) |
| 3 | \( 1 + 19.2T + 243T^{2} \) |
| 7 | \( 1 - 113.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 386.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.37e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 2.84e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 762.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.16e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 935.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.05e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 3.12e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 5.02e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.95e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.31e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.35e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.80e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.29e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.61e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.65e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.92e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.97e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.17e5T + 8.58e9T^{2} \) |
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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
−8.521624500441852856627890514711, −7.40653535959237442890390439366, −6.75124830891876003896781848104, −5.91867487765586319986587767568, −5.19612534811008213997573751209, −4.56643946684851987473593599618, −3.87855118758168951031871460170, −2.55932599385212572262657679586, −1.52867967532757177883889024760, 0,
1.52867967532757177883889024760, 2.55932599385212572262657679586, 3.87855118758168951031871460170, 4.56643946684851987473593599618, 5.19612534811008213997573751209, 5.91867487765586319986587767568, 6.75124830891876003896781848104, 7.40653535959237442890390439366, 8.521624500441852856627890514711
