| L(s) = 1 | − 5-s − 3·9-s + 11-s − 2·13-s + 6·17-s + 4·19-s + 4·23-s + 25-s − 6·29-s − 8·31-s + 2·37-s + 2·41-s − 4·43-s + 3·45-s − 12·47-s − 7·49-s + 2·53-s − 55-s − 4·59-s + 10·61-s + 2·65-s + 16·67-s + 8·71-s + 14·73-s + 8·79-s + 9·81-s + 4·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 9-s + 0.301·11-s − 0.554·13-s + 1.45·17-s + 0.917·19-s + 0.834·23-s + 1/5·25-s − 1.11·29-s − 1.43·31-s + 0.328·37-s + 0.312·41-s − 0.609·43-s + 0.447·45-s − 1.75·47-s − 49-s + 0.274·53-s − 0.134·55-s − 0.520·59-s + 1.28·61-s + 0.248·65-s + 1.95·67-s + 0.949·71-s + 1.63·73-s + 0.900·79-s + 81-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.454758861\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.454758861\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 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 | \( 1 - 10 T + p T^{2} \) |
| 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
−8.448904786013577857416673383279, −7.84109121338928955235594960129, −7.22566483951161934358497324239, −6.34700070707920035984611739420, −5.33894927898579958226937092107, −5.05872363288651094885006908150, −3.56946334592753091691191901249, −3.29934527379638532175228783242, −2.03779058571190597135714039078, −0.70358892174385003118951574686,
0.70358892174385003118951574686, 2.03779058571190597135714039078, 3.29934527379638532175228783242, 3.56946334592753091691191901249, 5.05872363288651094885006908150, 5.33894927898579958226937092107, 6.34700070707920035984611739420, 7.22566483951161934358497324239, 7.84109121338928955235594960129, 8.448904786013577857416673383279
