| L(s) = 1 | − 1.80·2-s + 3-s + 1.24·4-s − 1.44·5-s − 1.80·6-s − 3.44·7-s + 1.35·8-s + 9-s + 2.60·10-s + 5.18·11-s + 1.24·12-s + 6.20·14-s − 1.44·15-s − 4.93·16-s − 0.753·17-s − 1.80·18-s − 7.96·19-s − 1.80·20-s − 3.44·21-s − 9.34·22-s − 2.82·23-s + 1.35·24-s − 2.91·25-s + 27-s − 4.29·28-s − 3.91·29-s + 2.60·30-s + ⋯ |
| L(s) = 1 | − 1.27·2-s + 0.577·3-s + 0.623·4-s − 0.646·5-s − 0.735·6-s − 1.30·7-s + 0.479·8-s + 0.333·9-s + 0.823·10-s + 1.56·11-s + 0.359·12-s + 1.65·14-s − 0.373·15-s − 1.23·16-s − 0.182·17-s − 0.424·18-s − 1.82·19-s − 0.402·20-s − 0.751·21-s − 1.99·22-s − 0.589·23-s + 0.276·24-s − 0.582·25-s + 0.192·27-s − 0.811·28-s − 0.726·29-s + 0.475·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.80T + 2T^{2} \) |
| 5 | \( 1 + 1.44T + 5T^{2} \) |
| 7 | \( 1 + 3.44T + 7T^{2} \) |
| 11 | \( 1 - 5.18T + 11T^{2} \) |
| 17 | \( 1 + 0.753T + 17T^{2} \) |
| 19 | \( 1 + 7.96T + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 + 3.91T + 29T^{2} \) |
| 31 | \( 1 - 4.89T + 31T^{2} \) |
| 37 | \( 1 + 6.24T + 37T^{2} \) |
| 41 | \( 1 + 1.80T + 41T^{2} \) |
| 43 | \( 1 + 7.09T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 + 3.08T + 53T^{2} \) |
| 59 | \( 1 + 1.87T + 59T^{2} \) |
| 61 | \( 1 - 3.34T + 61T^{2} \) |
| 67 | \( 1 - 4.54T + 67T^{2} \) |
| 71 | \( 1 - 9.11T + 71T^{2} \) |
| 73 | \( 1 + 2.95T + 73T^{2} \) |
| 79 | \( 1 + 9.43T + 79T^{2} \) |
| 83 | \( 1 - 6.46T + 83T^{2} \) |
| 89 | \( 1 + 1.15T + 89T^{2} \) |
| 97 | \( 1 + 8.65T + 97T^{2} \) |
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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
−10.04622159425429106782801160423, −9.545318827431046671283032706021, −8.688382530171444118859210317137, −8.134379072855960489066306656933, −6.89865178120035937609479784441, −6.42142572689273810047465156245, −4.32191340047312199842370424717, −3.52848953311962123409972105103, −1.85376290648341535326764824068, 0,
1.85376290648341535326764824068, 3.52848953311962123409972105103, 4.32191340047312199842370424717, 6.42142572689273810047465156245, 6.89865178120035937609479784441, 8.134379072855960489066306656933, 8.688382530171444118859210317137, 9.545318827431046671283032706021, 10.04622159425429106782801160423
