| L(s) = 1 | + 2·2-s + 9·3-s − 28·4-s + 18·6-s − 148·7-s − 120·8-s + 81·9-s + 121·11-s − 252·12-s − 574·13-s − 296·14-s + 656·16-s + 722·17-s + 162·18-s + 2.16e3·19-s − 1.33e3·21-s + 242·22-s + 2.53e3·23-s − 1.08e3·24-s − 1.14e3·26-s + 729·27-s + 4.14e3·28-s + 4.65e3·29-s + 5.03e3·31-s + 5.15e3·32-s + 1.08e3·33-s + 1.44e3·34-s + ⋯ |
| L(s) = 1 | + 0.353·2-s + 0.577·3-s − 7/8·4-s + 0.204·6-s − 1.14·7-s − 0.662·8-s + 1/3·9-s + 0.301·11-s − 0.505·12-s − 0.942·13-s − 0.403·14-s + 0.640·16-s + 0.605·17-s + 0.117·18-s + 1.37·19-s − 0.659·21-s + 0.106·22-s + 0.999·23-s − 0.382·24-s − 0.333·26-s + 0.192·27-s + 0.998·28-s + 1.02·29-s + 0.940·31-s + 0.889·32-s + 0.174·33-s + 0.214·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - p^{2} T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - p^{2} T \) |
| good | 2 | \( 1 - p T + p^{5} T^{2} \) |
| 7 | \( 1 + 148 T + p^{5} T^{2} \) |
| 13 | \( 1 + 574 T + p^{5} T^{2} \) |
| 17 | \( 1 - 722 T + p^{5} T^{2} \) |
| 19 | \( 1 - 2160 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2536 T + p^{5} T^{2} \) |
| 29 | \( 1 - 4650 T + p^{5} T^{2} \) |
| 31 | \( 1 - 5032 T + p^{5} T^{2} \) |
| 37 | \( 1 + 8118 T + p^{5} T^{2} \) |
| 41 | \( 1 + 5138 T + p^{5} T^{2} \) |
| 43 | \( 1 + 8304 T + p^{5} T^{2} \) |
| 47 | \( 1 + 24728 T + p^{5} T^{2} \) |
| 53 | \( 1 - 28746 T + p^{5} T^{2} \) |
| 59 | \( 1 + 5860 T + p^{5} T^{2} \) |
| 61 | \( 1 + 53658 T + p^{5} T^{2} \) |
| 67 | \( 1 + 30908 T + p^{5} T^{2} \) |
| 71 | \( 1 + 69648 T + p^{5} T^{2} \) |
| 73 | \( 1 - 18446 T + p^{5} T^{2} \) |
| 79 | \( 1 + 25300 T + p^{5} T^{2} \) |
| 83 | \( 1 - 17556 T + p^{5} T^{2} \) |
| 89 | \( 1 - 132570 T + p^{5} T^{2} \) |
| 97 | \( 1 + 70658 T + p^{5} T^{2} \) |
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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
−9.197682202532357401071031225457, −8.327659884178006099251367147773, −7.32307257370165193193729118392, −6.45971264770610533260042302949, −5.32057580755005780232390476506, −4.55683242215275067792635277195, −3.29484330230589858544761496822, −2.98788915195665851803688427304, −1.17909073628278229336725681337, 0,
1.17909073628278229336725681337, 2.98788915195665851803688427304, 3.29484330230589858544761496822, 4.55683242215275067792635277195, 5.32057580755005780232390476506, 6.45971264770610533260042302949, 7.32307257370165193193729118392, 8.327659884178006099251367147773, 9.197682202532357401071031225457
