| L(s) = 1 | + 2·5-s − 7-s − 3·9-s − 4·11-s + 6·13-s − 2·17-s − 4·19-s + 23-s − 25-s − 2·29-s + 4·31-s − 2·35-s − 2·37-s − 6·41-s − 12·43-s − 6·45-s + 12·47-s + 49-s − 10·53-s − 8·55-s + 2·61-s + 3·63-s + 12·65-s − 12·67-s − 8·71-s − 14·73-s + 4·77-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.377·7-s − 9-s − 1.20·11-s + 1.66·13-s − 0.485·17-s − 0.917·19-s + 0.208·23-s − 1/5·25-s − 0.371·29-s + 0.718·31-s − 0.338·35-s − 0.328·37-s − 0.937·41-s − 1.82·43-s − 0.894·45-s + 1.75·47-s + 1/7·49-s − 1.37·53-s − 1.07·55-s + 0.256·61-s + 0.377·63-s + 1.48·65-s − 1.46·67-s − 0.949·71-s − 1.63·73-s + 0.455·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.629864757144697075761649249740, −7.936752832419798088583911309136, −6.76779125371997435118335878269, −6.03444355635435776505460708981, −5.64762925876769612266391960587, −4.63035623931147042799336871275, −3.45337937449096169115012234404, −2.66380121303710023121018194078, −1.67103103844683342010393944797, 0,
1.67103103844683342010393944797, 2.66380121303710023121018194078, 3.45337937449096169115012234404, 4.63035623931147042799336871275, 5.64762925876769612266391960587, 6.03444355635435776505460708981, 6.76779125371997435118335878269, 7.936752832419798088583911309136, 8.629864757144697075761649249740
