| L(s) = 1 | − 5-s − 2·7-s + 4·11-s − 6·13-s + 4·17-s + 4·19-s − 6·23-s + 25-s + 29-s + 2·35-s − 8·37-s + 2·41-s + 4·43-s + 4·47-s − 3·49-s + 2·53-s − 4·55-s − 8·59-s + 10·61-s + 6·65-s − 10·67-s + 8·71-s − 8·77-s + 8·79-s + 6·83-s − 4·85-s − 6·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s + 1.20·11-s − 1.66·13-s + 0.970·17-s + 0.917·19-s − 1.25·23-s + 1/5·25-s + 0.185·29-s + 0.338·35-s − 1.31·37-s + 0.312·41-s + 0.609·43-s + 0.583·47-s − 3/7·49-s + 0.274·53-s − 0.539·55-s − 1.04·59-s + 1.28·61-s + 0.744·65-s − 1.22·67-s + 0.949·71-s − 0.911·77-s + 0.900·79-s + 0.658·83-s − 0.433·85-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.373260807\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.373260807\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−8.088083699710393620705951236530, −7.40376064054788239128125112465, −6.91608044558078071309301932539, −6.07032600658004455460921957090, −5.32258447566160792398213499452, −4.44719121575233074558454918551, −3.65226700481618669934198159132, −3.00045546092868765699835197250, −1.89856523418779091680005516342, −0.62579465272769471064720293739,
0.62579465272769471064720293739, 1.89856523418779091680005516342, 3.00045546092868765699835197250, 3.65226700481618669934198159132, 4.44719121575233074558454918551, 5.32258447566160792398213499452, 6.07032600658004455460921957090, 6.91608044558078071309301932539, 7.40376064054788239128125112465, 8.088083699710393620705951236530
