| L(s) = 1 | − 2.99·3-s + 0.369·5-s + 7-s + 5.96·9-s − 3.93·11-s + 6.07·13-s − 1.10·15-s + 6.75·17-s − 0.117·19-s − 2.99·21-s + 8.67·23-s − 4.86·25-s − 8.86·27-s − 6.27·29-s − 0.808·31-s + 11.7·33-s + 0.369·35-s + 10.1·37-s − 18.1·39-s − 7.27·41-s + 0.0365·43-s + 2.20·45-s + 4.06·47-s + 49-s − 20.2·51-s + 2.53·53-s − 1.45·55-s + ⋯ |
| L(s) = 1 | − 1.72·3-s + 0.165·5-s + 0.377·7-s + 1.98·9-s − 1.18·11-s + 1.68·13-s − 0.285·15-s + 1.63·17-s − 0.0268·19-s − 0.653·21-s + 1.80·23-s − 0.972·25-s − 1.70·27-s − 1.16·29-s − 0.145·31-s + 2.05·33-s + 0.0625·35-s + 1.67·37-s − 2.91·39-s − 1.13·41-s + 0.00557·43-s + 0.328·45-s + 0.593·47-s + 0.142·49-s − 2.83·51-s + 0.348·53-s − 0.196·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.157420989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.157420989\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + 2.99T + 3T^{2} \) |
| 5 | \( 1 - 0.369T + 5T^{2} \) |
| 11 | \( 1 + 3.93T + 11T^{2} \) |
| 13 | \( 1 - 6.07T + 13T^{2} \) |
| 17 | \( 1 - 6.75T + 17T^{2} \) |
| 19 | \( 1 + 0.117T + 19T^{2} \) |
| 23 | \( 1 - 8.67T + 23T^{2} \) |
| 29 | \( 1 + 6.27T + 29T^{2} \) |
| 31 | \( 1 + 0.808T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 + 7.27T + 41T^{2} \) |
| 43 | \( 1 - 0.0365T + 43T^{2} \) |
| 47 | \( 1 - 4.06T + 47T^{2} \) |
| 53 | \( 1 - 2.53T + 53T^{2} \) |
| 59 | \( 1 - 5.53T + 59T^{2} \) |
| 61 | \( 1 + 6.02T + 61T^{2} \) |
| 67 | \( 1 + 4.88T + 67T^{2} \) |
| 71 | \( 1 - 2.35T + 71T^{2} \) |
| 73 | \( 1 + 6.82T + 73T^{2} \) |
| 79 | \( 1 - 6.35T + 79T^{2} \) |
| 83 | \( 1 + 8.65T + 83T^{2} \) |
| 89 | \( 1 + 3.56T + 89T^{2} \) |
| 97 | \( 1 + 0.211T + 97T^{2} \) |
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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
−8.429551212632253735624932572474, −7.65763879906127589228937529937, −7.01396367247559903261803411101, −5.94826082069345483166993160563, −5.67376793252090744547341865134, −5.06564008791129246618427844198, −4.11319260033177625008588938623, −3.11550490017169597440693486884, −1.57997852915385340810846707820, −0.74560308170913616628124044394,
0.74560308170913616628124044394, 1.57997852915385340810846707820, 3.11550490017169597440693486884, 4.11319260033177625008588938623, 5.06564008791129246618427844198, 5.67376793252090744547341865134, 5.94826082069345483166993160563, 7.01396367247559903261803411101, 7.65763879906127589228937529937, 8.429551212632253735624932572474
