| L(s) = 1 | + 2·5-s + 2·7-s − 4·11-s − 13-s + 6·19-s + 4·23-s − 25-s + 8·29-s + 2·31-s + 4·35-s + 6·37-s − 6·41-s + 8·43-s + 8·47-s − 3·49-s − 12·53-s − 8·55-s + 4·59-s + 10·61-s − 2·65-s + 2·67-s − 16·71-s + 14·73-s − 8·77-s + 4·79-s − 12·83-s + 6·89-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.755·7-s − 1.20·11-s − 0.277·13-s + 1.37·19-s + 0.834·23-s − 1/5·25-s + 1.48·29-s + 0.359·31-s + 0.676·35-s + 0.986·37-s − 0.937·41-s + 1.21·43-s + 1.16·47-s − 3/7·49-s − 1.64·53-s − 1.07·55-s + 0.520·59-s + 1.28·61-s − 0.248·65-s + 0.244·67-s − 1.89·71-s + 1.63·73-s − 0.911·77-s + 0.450·79-s − 1.31·83-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.215696042\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.215696042\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−9.374800673600871396369360815682, −8.345885893689355266865433880677, −7.74982624300516502645234312538, −6.91437440844756442579529153190, −5.85428751909866229664883723997, −5.21397578582522565243253635444, −4.55774581547134998480701833621, −3.06816752901810724197585330945, −2.30205596173135729346442665808, −1.06736363009885798558856425967,
1.06736363009885798558856425967, 2.30205596173135729346442665808, 3.06816752901810724197585330945, 4.55774581547134998480701833621, 5.21397578582522565243253635444, 5.85428751909866229664883723997, 6.91437440844756442579529153190, 7.74982624300516502645234312538, 8.345885893689355266865433880677, 9.374800673600871396369360815682
