| L(s) = 1 | − 5-s − 4.11·7-s + 2.11·11-s + 4.80·13-s − 4·17-s + 19-s + 0.691·23-s + 25-s − 4.11·29-s − 2·31-s + 4.11·35-s + 3.42·37-s − 5.42·41-s − 2.80·43-s − 0.691·47-s + 9.91·49-s + 2.69·53-s − 2.11·55-s − 9.53·59-s − 2.91·61-s − 4.80·65-s + 4·67-s − 5.60·71-s + 6·73-s − 8.69·77-s + 15.1·79-s − 6.22·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.55·7-s + 0.637·11-s + 1.33·13-s − 0.970·17-s + 0.229·19-s + 0.144·23-s + 0.200·25-s − 0.763·29-s − 0.359·31-s + 0.695·35-s + 0.562·37-s − 0.846·41-s − 0.427·43-s − 0.100·47-s + 1.41·49-s + 0.369·53-s − 0.284·55-s − 1.24·59-s − 0.373·61-s − 0.595·65-s + 0.488·67-s − 0.665·71-s + 0.702·73-s − 0.990·77-s + 1.70·79-s − 0.683·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.221493029\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.221493029\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + 4.11T + 7T^{2} \) |
| 11 | \( 1 - 2.11T + 11T^{2} \) |
| 13 | \( 1 - 4.80T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 23 | \( 1 - 0.691T + 23T^{2} \) |
| 29 | \( 1 + 4.11T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 3.42T + 37T^{2} \) |
| 41 | \( 1 + 5.42T + 41T^{2} \) |
| 43 | \( 1 + 2.80T + 43T^{2} \) |
| 47 | \( 1 + 0.691T + 47T^{2} \) |
| 53 | \( 1 - 2.69T + 53T^{2} \) |
| 59 | \( 1 + 9.53T + 59T^{2} \) |
| 61 | \( 1 + 2.91T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 5.60T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 + 6.22T + 83T^{2} \) |
| 89 | \( 1 - 1.42T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−7.943936601719916495226081008627, −7.15662901913473291712496317273, −6.38942274588562971702310613518, −6.22242969175495081563243247493, −5.15393651104344861543434160364, −4.10689326046625969273427254169, −3.60727152806803622941416807047, −2.97391527214936521217148963772, −1.77249081805890786946030085571, −0.55982416789551528294546411912,
0.55982416789551528294546411912, 1.77249081805890786946030085571, 2.97391527214936521217148963772, 3.60727152806803622941416807047, 4.10689326046625969273427254169, 5.15393651104344861543434160364, 6.22242969175495081563243247493, 6.38942274588562971702310613518, 7.15662901913473291712496317273, 7.943936601719916495226081008627
