| L(s) = 1 | + 2·3-s + 4·7-s + 9-s + 11-s + 5·13-s + 7·19-s + 8·21-s − 3·23-s − 4·27-s + 3·29-s − 5·31-s + 2·33-s − 4·37-s + 10·39-s + 12·41-s − 5·43-s + 9·49-s + 6·53-s + 14·57-s − 12·59-s − 10·61-s + 4·63-s − 14·67-s − 6·69-s − 3·71-s + 8·73-s + 4·77-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.51·7-s + 1/3·9-s + 0.301·11-s + 1.38·13-s + 1.60·19-s + 1.74·21-s − 0.625·23-s − 0.769·27-s + 0.557·29-s − 0.898·31-s + 0.348·33-s − 0.657·37-s + 1.60·39-s + 1.87·41-s − 0.762·43-s + 9/7·49-s + 0.824·53-s + 1.85·57-s − 1.56·59-s − 1.28·61-s + 0.503·63-s − 1.71·67-s − 0.722·69-s − 0.356·71-s + 0.936·73-s + 0.455·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.988831941\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.988831941\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−8.215066690317924879429209850018, −7.897985588647675963820965641286, −7.23198558052927413132140922600, −6.07659462756156825046100662425, −5.40373239996389746725721759481, −4.46571875991929313351266021179, −3.69956159581703457314506928919, −2.97493428037670400673680749715, −1.87861636891240765312856224651, −1.21298979508838572831757993102,
1.21298979508838572831757993102, 1.87861636891240765312856224651, 2.97493428037670400673680749715, 3.69956159581703457314506928919, 4.46571875991929313351266021179, 5.40373239996389746725721759481, 6.07659462756156825046100662425, 7.23198558052927413132140922600, 7.897985588647675963820965641286, 8.215066690317924879429209850018
