| L(s) = 1 | + 3-s + 2·7-s + 9-s + 11-s + 13-s − 2·17-s − 6·19-s + 2·21-s − 8·23-s − 5·25-s + 27-s − 2·29-s + 6·31-s + 33-s − 6·37-s + 39-s − 8·41-s + 8·43-s − 12·47-s − 3·49-s − 2·51-s − 10·53-s − 6·57-s + 8·59-s + 2·61-s + 2·63-s − 6·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.755·7-s + 1/3·9-s + 0.301·11-s + 0.277·13-s − 0.485·17-s − 1.37·19-s + 0.436·21-s − 1.66·23-s − 25-s + 0.192·27-s − 0.371·29-s + 1.07·31-s + 0.174·33-s − 0.986·37-s + 0.160·39-s − 1.24·41-s + 1.21·43-s − 1.75·47-s − 3/7·49-s − 0.280·51-s − 1.37·53-s − 0.794·57-s + 1.04·59-s + 0.256·61-s + 0.251·63-s − 0.733·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−7.896346536673545375228039523964, −6.87629988840599459978918018319, −6.31511090298148607783456243488, −5.51842665157610368102255930696, −4.48034192655992479210652663082, −4.11876322993667465169771384712, −3.19127765612886888482751695381, −2.06478507097045624057450072916, −1.65319622876630168091099596040, 0,
1.65319622876630168091099596040, 2.06478507097045624057450072916, 3.19127765612886888482751695381, 4.11876322993667465169771384712, 4.48034192655992479210652663082, 5.51842665157610368102255930696, 6.31511090298148607783456243488, 6.87629988840599459978918018319, 7.896346536673545375228039523964
