| L(s) = 1 | + 0.866·5-s − 3.51·11-s + 1.86·13-s + 6.51·17-s + 5.38·19-s + 8.64·23-s − 4.24·25-s + 3.51·29-s + 1.86·31-s − 2.78·37-s − 10.3·41-s − 5.78·43-s − 6.16·47-s + 5.60·53-s − 3.04·55-s + 5.64·59-s − 10.2·61-s + 1.61·65-s − 1.35·67-s + 2.08·71-s − 7.24·73-s + 11.6·79-s − 6.86·83-s + 5.64·85-s + 6.56·89-s + 4.66·95-s + 3.29·97-s + ⋯ |
| L(s) = 1 | + 0.387·5-s − 1.05·11-s + 0.517·13-s + 1.58·17-s + 1.23·19-s + 1.80·23-s − 0.849·25-s + 0.652·29-s + 0.335·31-s − 0.457·37-s − 1.62·41-s − 0.881·43-s − 0.898·47-s + 0.769·53-s − 0.410·55-s + 0.735·59-s − 1.31·61-s + 0.200·65-s − 0.165·67-s + 0.247·71-s − 0.848·73-s + 1.31·79-s − 0.753·83-s + 0.612·85-s + 0.695·89-s + 0.478·95-s + 0.334·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.253317998\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.253317998\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 0.866T + 5T^{2} \) |
| 11 | \( 1 + 3.51T + 11T^{2} \) |
| 13 | \( 1 - 1.86T + 13T^{2} \) |
| 17 | \( 1 - 6.51T + 17T^{2} \) |
| 19 | \( 1 - 5.38T + 19T^{2} \) |
| 23 | \( 1 - 8.64T + 23T^{2} \) |
| 29 | \( 1 - 3.51T + 29T^{2} \) |
| 31 | \( 1 - 1.86T + 31T^{2} \) |
| 37 | \( 1 + 2.78T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 + 5.78T + 43T^{2} \) |
| 47 | \( 1 + 6.16T + 47T^{2} \) |
| 53 | \( 1 - 5.60T + 53T^{2} \) |
| 59 | \( 1 - 5.64T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 + 1.35T + 67T^{2} \) |
| 71 | \( 1 - 2.08T + 71T^{2} \) |
| 73 | \( 1 + 7.24T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 + 6.86T + 83T^{2} \) |
| 89 | \( 1 - 6.56T + 89T^{2} \) |
| 97 | \( 1 - 3.29T + 97T^{2} \) |
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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
−8.165551987062118743821717010805, −7.50071597157167370355764207900, −6.84411379730044903236320143882, −5.88459709108867830439070547459, −5.29157791171387491527959074883, −4.77911027213758305884644095089, −3.31652597542035778098847736035, −3.12639988166692634949231897002, −1.80668841929584293344570705981, −0.842541393332245055864221350362,
0.842541393332245055864221350362, 1.80668841929584293344570705981, 3.12639988166692634949231897002, 3.31652597542035778098847736035, 4.77911027213758305884644095089, 5.29157791171387491527959074883, 5.88459709108867830439070547459, 6.84411379730044903236320143882, 7.50071597157167370355764207900, 8.165551987062118743821717010805
