| L(s) = 1 | + 9·3-s + 21·5-s + 143·7-s + 81·9-s + 205·11-s − 78·13-s + 189·15-s − 2.12e3·17-s − 361·19-s + 1.28e3·21-s − 20·23-s − 2.68e3·25-s + 729·27-s − 4.86e3·29-s + 1.09e3·31-s + 1.84e3·33-s + 3.00e3·35-s − 1.51e4·37-s − 702·39-s − 9.40e3·41-s − 2.00e4·43-s + 1.70e3·45-s − 1.41e4·47-s + 3.64e3·49-s − 1.91e4·51-s + 2.63e4·53-s + 4.30e3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.375·5-s + 1.10·7-s + 1/3·9-s + 0.510·11-s − 0.128·13-s + 0.216·15-s − 1.78·17-s − 0.229·19-s + 0.636·21-s − 0.00788·23-s − 0.858·25-s + 0.192·27-s − 1.07·29-s + 0.205·31-s + 0.294·33-s + 0.414·35-s − 1.81·37-s − 0.0739·39-s − 0.873·41-s − 1.65·43-s + 0.125·45-s − 0.931·47-s + 0.216·49-s − 1.02·51-s + 1.29·53-s + 0.191·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 - p^{2} T \) |
| 19 | \( 1 + p^{2} T \) |
| good | 5 | \( 1 - 21 T + p^{5} T^{2} \) |
| 7 | \( 1 - 143 T + p^{5} T^{2} \) |
| 11 | \( 1 - 205 T + p^{5} T^{2} \) |
| 13 | \( 1 + 6 p T + p^{5} T^{2} \) |
| 17 | \( 1 + 125 p T + p^{5} T^{2} \) |
| 23 | \( 1 + 20 T + p^{5} T^{2} \) |
| 29 | \( 1 + 4866 T + p^{5} T^{2} \) |
| 31 | \( 1 - 1098 T + p^{5} T^{2} \) |
| 37 | \( 1 + 15128 T + p^{5} T^{2} \) |
| 41 | \( 1 + 9400 T + p^{5} T^{2} \) |
| 43 | \( 1 + 20073 T + p^{5} T^{2} \) |
| 47 | \( 1 + 14105 T + p^{5} T^{2} \) |
| 53 | \( 1 - 26386 T + p^{5} T^{2} \) |
| 59 | \( 1 - 224 p T + p^{5} T^{2} \) |
| 61 | \( 1 + 2293 T + p^{5} T^{2} \) |
| 67 | \( 1 + 35976 T + p^{5} T^{2} \) |
| 71 | \( 1 + 10180 T + p^{5} T^{2} \) |
| 73 | \( 1 - 33109 T + p^{5} T^{2} \) |
| 79 | \( 1 - 53888 T + p^{5} T^{2} \) |
| 83 | \( 1 + 75196 T + p^{5} T^{2} \) |
| 89 | \( 1 - 20618 T + p^{5} T^{2} \) |
| 97 | \( 1 + 84130 T + p^{5} T^{2} \) |
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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
−8.768064587610505525985687627305, −8.328967813110492470341723779655, −7.23773896933206521632003999963, −6.50154267500784974240945861554, −5.29057865398867198267630665857, −4.47526876980078647648944019143, −3.52514115499864810731496895552, −2.09154026254166439186745325886, −1.66622236771914193776676946226, 0,
1.66622236771914193776676946226, 2.09154026254166439186745325886, 3.52514115499864810731496895552, 4.47526876980078647648944019143, 5.29057865398867198267630665857, 6.50154267500784974240945861554, 7.23773896933206521632003999963, 8.328967813110492470341723779655, 8.768064587610505525985687627305
