| L(s) = 1 | + 3-s − 2·7-s + 9-s − 11-s − 2·13-s − 2·19-s − 2·21-s + 23-s − 5·25-s + 27-s + 10·29-s + 4·31-s − 33-s − 2·37-s − 2·39-s − 2·41-s − 2·43-s − 8·47-s − 3·49-s + 4·53-s − 2·57-s + 12·59-s + 6·61-s − 2·63-s − 2·67-s + 69-s − 6·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s − 0.458·19-s − 0.436·21-s + 0.208·23-s − 25-s + 0.192·27-s + 1.85·29-s + 0.718·31-s − 0.174·33-s − 0.328·37-s − 0.320·39-s − 0.312·41-s − 0.304·43-s − 1.16·47-s − 3/7·49-s + 0.549·53-s − 0.264·57-s + 1.56·59-s + 0.768·61-s − 0.251·63-s − 0.244·67-s + 0.120·69-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 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
−14.79398978821930, −14.25970866960777, −13.82899962835067, −13.16237311134513, −12.96787559737562, −12.28015913451469, −11.79188531301053, −11.30742536399333, −10.38715879519555, −10.04282170957543, −9.799558313121192, −9.028189338724716, −8.391201038558696, −8.184038695556983, −7.342298713253379, −6.864682919643963, −6.358511844065854, −5.741057424540351, −4.910878078823751, −4.506025515198216, −3.689288810202044, −3.149647249238811, −2.540368939108451, −1.947435800635181, −0.9341388421155130, 0,
0.9341388421155130, 1.947435800635181, 2.540368939108451, 3.149647249238811, 3.689288810202044, 4.506025515198216, 4.910878078823751, 5.741057424540351, 6.358511844065854, 6.864682919643963, 7.342298713253379, 8.184038695556983, 8.391201038558696, 9.028189338724716, 9.799558313121192, 10.04282170957543, 10.38715879519555, 11.30742536399333, 11.79188531301053, 12.28015913451469, 12.96787559737562, 13.16237311134513, 13.82899962835067, 14.25970866960777, 14.79398978821930
