| L(s) = 1 | − 7-s + 4·11-s − 3·13-s + 7·17-s − 6·19-s − 9·23-s − 3·29-s + 7·31-s − 10·37-s − 41-s − 13·43-s + 2·47-s + 49-s + 53-s − 11·59-s − 13·61-s − 8·71-s − 8·73-s − 4·77-s − 4·79-s + 7·83-s − 14·89-s + 3·91-s − 8·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 1.20·11-s − 0.832·13-s + 1.69·17-s − 1.37·19-s − 1.87·23-s − 0.557·29-s + 1.25·31-s − 1.64·37-s − 0.156·41-s − 1.98·43-s + 0.291·47-s + 1/7·49-s + 0.137·53-s − 1.43·59-s − 1.66·61-s − 0.949·71-s − 0.936·73-s − 0.455·77-s − 0.450·79-s + 0.768·83-s − 1.48·89-s + 0.314·91-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8394516501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8394516501\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 + 13 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.83275145919606, −13.38395931841878, −12.50769661130856, −12.18539753242353, −12.03791630862153, −11.47746439984465, −10.61395803904100, −10.24950931261517, −9.818529550511175, −9.459442026590839, −8.647305352610609, −8.378660515515713, −7.711306580641242, −7.215720063217799, −6.611911025188266, −6.120823480457950, −5.754782568079645, −4.949060100825217, −4.405198028640253, −3.839575646220533, −3.315041468689279, −2.702894299648950, −1.728111086616192, −1.514023488467416, −0.2757708946431827,
0.2757708946431827, 1.514023488467416, 1.728111086616192, 2.702894299648950, 3.315041468689279, 3.839575646220533, 4.405198028640253, 4.949060100825217, 5.754782568079645, 6.120823480457950, 6.611911025188266, 7.215720063217799, 7.711306580641242, 8.378660515515713, 8.647305352610609, 9.459442026590839, 9.818529550511175, 10.24950931261517, 10.61395803904100, 11.47746439984465, 12.03791630862153, 12.18539753242353, 12.50769661130856, 13.38395931841878, 13.83275145919606
