| L(s) = 1 | + 2·7-s − 2·13-s − 6·17-s + 4·19-s − 6·23-s − 6·29-s + 4·31-s − 2·37-s − 6·41-s − 10·43-s + 6·47-s − 3·49-s − 6·53-s + 12·59-s + 2·61-s + 2·67-s − 12·71-s − 2·73-s − 8·79-s − 6·83-s + 6·89-s − 4·91-s − 2·97-s − 6·101-s + 14·103-s + 6·107-s + 2·109-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 0.554·13-s − 1.45·17-s + 0.917·19-s − 1.25·23-s − 1.11·29-s + 0.718·31-s − 0.328·37-s − 0.937·41-s − 1.52·43-s + 0.875·47-s − 3/7·49-s − 0.824·53-s + 1.56·59-s + 0.256·61-s + 0.244·67-s − 1.42·71-s − 0.234·73-s − 0.900·79-s − 0.658·83-s + 0.635·89-s − 0.419·91-s − 0.203·97-s − 0.597·101-s + 1.37·103-s + 0.580·107-s + 0.191·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−8.239614848469987967095910594928, −7.44058971815937736168226950763, −6.80259195280943619843400752261, −5.89044182431317279877326881053, −5.06622402809054522758907056720, −4.43681566700912263057071092700, −3.51423251581280431265052128819, −2.37574826007780752457165155100, −1.58920274451173242913221261639, 0,
1.58920274451173242913221261639, 2.37574826007780752457165155100, 3.51423251581280431265052128819, 4.43681566700912263057071092700, 5.06622402809054522758907056720, 5.89044182431317279877326881053, 6.80259195280943619843400752261, 7.44058971815937736168226950763, 8.239614848469987967095910594928
