| L(s) = 1 | + 2·3-s + 4·7-s + 9-s + 2·11-s + 2·13-s + 2·17-s − 2·19-s + 8·21-s − 4·23-s − 4·27-s + 6·29-s + 4·33-s + 10·37-s + 4·39-s − 6·41-s + 6·43-s + 8·47-s + 9·49-s + 4·51-s − 6·53-s − 4·57-s − 14·59-s − 2·61-s + 4·63-s + 10·67-s − 8·69-s + 12·71-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.51·7-s + 1/3·9-s + 0.603·11-s + 0.554·13-s + 0.485·17-s − 0.458·19-s + 1.74·21-s − 0.834·23-s − 0.769·27-s + 1.11·29-s + 0.696·33-s + 1.64·37-s + 0.640·39-s − 0.937·41-s + 0.914·43-s + 1.16·47-s + 9/7·49-s + 0.560·51-s − 0.824·53-s − 0.529·57-s − 1.82·59-s − 0.256·61-s + 0.503·63-s + 1.22·67-s − 0.963·69-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.610321058\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.610321058\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−8.574936309637445379610258972840, −7.999985048107733082505585438448, −7.57608379011211040426322201256, −6.41693218482401941386600750161, −5.64174311222249935302347088206, −4.55273331792869159699742135970, −4.01097757147887956643150593139, −2.97994173424545511695184613179, −2.07247521067156947562182835303, −1.21396611795770319909404297795,
1.21396611795770319909404297795, 2.07247521067156947562182835303, 2.97994173424545511695184613179, 4.01097757147887956643150593139, 4.55273331792869159699742135970, 5.64174311222249935302347088206, 6.41693218482401941386600750161, 7.57608379011211040426322201256, 7.999985048107733082505585438448, 8.574936309637445379610258972840
