| L(s) = 1 | − 2.63·2-s − 3-s + 4.95·4-s − 0.496·5-s + 2.63·6-s + 7-s − 7.80·8-s + 9-s + 1.30·10-s + 2.94·11-s − 4.95·12-s + 13-s − 2.63·14-s + 0.496·15-s + 10.6·16-s + 17-s − 2.63·18-s + 3.33·19-s − 2.46·20-s − 21-s − 7.76·22-s − 4.96·23-s + 7.80·24-s − 4.75·25-s − 2.63·26-s − 27-s + 4.95·28-s + ⋯ |
| L(s) = 1 | − 1.86·2-s − 0.577·3-s + 2.47·4-s − 0.221·5-s + 1.07·6-s + 0.377·7-s − 2.75·8-s + 0.333·9-s + 0.413·10-s + 0.887·11-s − 1.43·12-s + 0.277·13-s − 0.705·14-s + 0.128·15-s + 2.66·16-s + 0.242·17-s − 0.621·18-s + 0.765·19-s − 0.550·20-s − 0.218·21-s − 1.65·22-s − 1.03·23-s + 1.59·24-s − 0.950·25-s − 0.517·26-s − 0.192·27-s + 0.937·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4641 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4641 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7054441217\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7054441217\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + 2.63T + 2T^{2} \) |
| 5 | \( 1 + 0.496T + 5T^{2} \) |
| 11 | \( 1 - 2.94T + 11T^{2} \) |
| 19 | \( 1 - 3.33T + 19T^{2} \) |
| 23 | \( 1 + 4.96T + 23T^{2} \) |
| 29 | \( 1 - 6.05T + 29T^{2} \) |
| 31 | \( 1 - 5.44T + 31T^{2} \) |
| 37 | \( 1 - 2.79T + 37T^{2} \) |
| 41 | \( 1 + 2.20T + 41T^{2} \) |
| 43 | \( 1 - 6.80T + 43T^{2} \) |
| 47 | \( 1 - 4.20T + 47T^{2} \) |
| 53 | \( 1 + 2.03T + 53T^{2} \) |
| 59 | \( 1 + 9.59T + 59T^{2} \) |
| 61 | \( 1 - 9.95T + 61T^{2} \) |
| 67 | \( 1 - 8.02T + 67T^{2} \) |
| 71 | \( 1 + 15.0T + 71T^{2} \) |
| 73 | \( 1 + 1.91T + 73T^{2} \) |
| 79 | \( 1 - 10.6T + 79T^{2} \) |
| 83 | \( 1 + 2.93T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−8.298044974859031350241518236573, −7.75459921733093850716444822950, −7.12460525191693490873050036132, −6.28337639685910978078867421253, −5.87206099707665349482419460624, −4.60866560349916346771907292052, −3.60194245321310313368699164272, −2.43761401387738604452755086910, −1.44407977619262601529035281359, −0.69628721471548350415828270310,
0.69628721471548350415828270310, 1.44407977619262601529035281359, 2.43761401387738604452755086910, 3.60194245321310313368699164272, 4.60866560349916346771907292052, 5.87206099707665349482419460624, 6.28337639685910978078867421253, 7.12460525191693490873050036132, 7.75459921733093850716444822950, 8.298044974859031350241518236573
