| L(s) = 1 | + 3-s − 2·9-s + 5·11-s + 7·13-s + 3·17-s + 2·19-s − 8·23-s − 5·27-s + 5·29-s − 10·31-s + 5·33-s + 4·37-s + 7·39-s − 6·41-s + 2·43-s + 7·47-s + 3·51-s − 10·53-s + 2·57-s + 10·59-s + 12·61-s − 2·67-s − 8·69-s + 2·73-s − 7·79-s + 81-s + 4·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2/3·9-s + 1.50·11-s + 1.94·13-s + 0.727·17-s + 0.458·19-s − 1.66·23-s − 0.962·27-s + 0.928·29-s − 1.79·31-s + 0.870·33-s + 0.657·37-s + 1.12·39-s − 0.937·41-s + 0.304·43-s + 1.02·47-s + 0.420·51-s − 1.37·53-s + 0.264·57-s + 1.30·59-s + 1.53·61-s − 0.244·67-s − 0.963·69-s + 0.234·73-s − 0.787·79-s + 1/9·81-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.769544438\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.769544438\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−14.14061397084166, −13.65081155144564, −13.19540068695141, −12.44154477699505, −12.01841203972922, −11.48390296032573, −11.14576659483409, −10.58826165657329, −9.755913873107077, −9.533407530383171, −8.835587396603871, −8.458095973490002, −8.134257566207541, −7.433089204199754, −6.739405058346448, −6.255072498030034, −5.721080103446356, −5.382812554498972, −4.161291727405895, −3.914600125132963, −3.451163595089149, −2.824607841638348, −1.879340274803813, −1.422497259799154, −0.6332490383571889,
0.6332490383571889, 1.422497259799154, 1.879340274803813, 2.824607841638348, 3.451163595089149, 3.914600125132963, 4.161291727405895, 5.382812554498972, 5.721080103446356, 6.255072498030034, 6.739405058346448, 7.433089204199754, 8.134257566207541, 8.458095973490002, 8.835587396603871, 9.533407530383171, 9.755913873107077, 10.58826165657329, 11.14576659483409, 11.48390296032573, 12.01841203972922, 12.44154477699505, 13.19540068695141, 13.65081155144564, 14.14061397084166
