| L(s) = 1 | − 2·2-s + 3-s + 2·4-s + 5-s − 2·6-s − 2·7-s + 9-s − 2·10-s − 3·11-s + 2·12-s − 13-s + 4·14-s + 15-s − 4·16-s + 3·17-s − 2·18-s − 5·19-s + 2·20-s − 2·21-s + 6·22-s − 6·23-s + 25-s + 2·26-s + 27-s − 4·28-s − 2·30-s − 8·31-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s − 0.755·7-s + 1/3·9-s − 0.632·10-s − 0.904·11-s + 0.577·12-s − 0.277·13-s + 1.06·14-s + 0.258·15-s − 16-s + 0.727·17-s − 0.471·18-s − 1.14·19-s + 0.447·20-s − 0.436·21-s + 1.27·22-s − 1.25·23-s + 1/5·25-s + 0.392·26-s + 0.192·27-s − 0.755·28-s − 0.365·30-s − 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 885 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 885 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.697303928270612343468903921998, −8.945945465031174997582292318085, −8.161313789220962882239366440642, −7.49361399678913815522805343050, −6.58412004907401727467528669745, −5.51253295860538873867145864035, −4.11446641573101929177044682093, −2.76071489868874288514956060985, −1.78879436713783491506035462654, 0,
1.78879436713783491506035462654, 2.76071489868874288514956060985, 4.11446641573101929177044682093, 5.51253295860538873867145864035, 6.58412004907401727467528669745, 7.49361399678913815522805343050, 8.161313789220962882239366440642, 8.945945465031174997582292318085, 9.697303928270612343468903921998
