| L(s) = 1 | + 3-s − 5-s + 5·7-s − 26·9-s + 10·11-s − 13·13-s − 15-s + 93·17-s − 82·19-s + 5·21-s − 192·23-s − 124·25-s − 53·27-s − 106·29-s + 172·31-s + 10·33-s − 5·35-s + 379·37-s − 13·39-s − 148·41-s − 329·43-s + 26·45-s − 631·47-s − 318·49-s + 93·51-s + 160·53-s − 10·55-s + ⋯ |
| L(s) = 1 | + 0.192·3-s − 0.0894·5-s + 0.269·7-s − 0.962·9-s + 0.274·11-s − 0.277·13-s − 0.0172·15-s + 1.32·17-s − 0.990·19-s + 0.0519·21-s − 1.74·23-s − 0.991·25-s − 0.377·27-s − 0.678·29-s + 0.996·31-s + 0.0527·33-s − 0.0241·35-s + 1.68·37-s − 0.0533·39-s − 0.563·41-s − 1.16·43-s + 0.0861·45-s − 1.95·47-s − 0.927·49-s + 0.255·51-s + 0.414·53-s − 0.0245·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + p T \) |
| good | 3 | \( 1 - T + p^{3} T^{2} \) |
| 5 | \( 1 + T + p^{3} T^{2} \) |
| 7 | \( 1 - 5 T + p^{3} T^{2} \) |
| 11 | \( 1 - 10 T + p^{3} T^{2} \) |
| 17 | \( 1 - 93 T + p^{3} T^{2} \) |
| 19 | \( 1 + 82 T + p^{3} T^{2} \) |
| 23 | \( 1 + 192 T + p^{3} T^{2} \) |
| 29 | \( 1 + 106 T + p^{3} T^{2} \) |
| 31 | \( 1 - 172 T + p^{3} T^{2} \) |
| 37 | \( 1 - 379 T + p^{3} T^{2} \) |
| 41 | \( 1 + 148 T + p^{3} T^{2} \) |
| 43 | \( 1 + 329 T + p^{3} T^{2} \) |
| 47 | \( 1 + 631 T + p^{3} T^{2} \) |
| 53 | \( 1 - 160 T + p^{3} T^{2} \) |
| 59 | \( 1 + 478 T + p^{3} T^{2} \) |
| 61 | \( 1 - 300 T + p^{3} T^{2} \) |
| 67 | \( 1 + 722 T + p^{3} T^{2} \) |
| 71 | \( 1 - 335 T + p^{3} T^{2} \) |
| 73 | \( 1 - 90 T + p^{3} T^{2} \) |
| 79 | \( 1 + 788 T + p^{3} T^{2} \) |
| 83 | \( 1 - 96 T + p^{3} T^{2} \) |
| 89 | \( 1 + 866 T + p^{3} T^{2} \) |
| 97 | \( 1 + 998 T + p^{3} T^{2} \) |
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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
−10.20046068172938889162196910891, −9.539285530984600710001055192192, −8.214198244785336280501904603421, −7.930640773995339376148046080683, −6.40857065564106316434350429548, −5.61613784608317344015897061709, −4.32945773792707444165859889356, −3.17442755373337081861436461936, −1.84086539159772154942608010769, 0,
1.84086539159772154942608010769, 3.17442755373337081861436461936, 4.32945773792707444165859889356, 5.61613784608317344015897061709, 6.40857065564106316434350429548, 7.930640773995339376148046080683, 8.214198244785336280501904603421, 9.539285530984600710001055192192, 10.20046068172938889162196910891
