| L(s) = 1 | + 0.732·2-s − 1.46·4-s − 5-s − 4.46·7-s − 2.53·8-s − 0.732·10-s − 3.46·11-s − 3.26·14-s + 1.07·16-s + 6.73·17-s + 5.46·19-s + 1.46·20-s − 2.53·22-s + 0.535·23-s + 25-s + 6.53·28-s + 2.73·29-s + 3.19·31-s + 5.85·32-s + 4.92·34-s + 4.46·35-s − 4·37-s + 4·38-s + 2.53·40-s − 5.26·41-s − 0.267·43-s + 5.07·44-s + ⋯ |
| L(s) = 1 | + 0.517·2-s − 0.732·4-s − 0.447·5-s − 1.68·7-s − 0.896·8-s − 0.231·10-s − 1.04·11-s − 0.873·14-s + 0.267·16-s + 1.63·17-s + 1.25·19-s + 0.327·20-s − 0.540·22-s + 0.111·23-s + 0.200·25-s + 1.23·28-s + 0.507·29-s + 0.574·31-s + 1.03·32-s + 0.845·34-s + 0.754·35-s − 0.657·37-s + 0.648·38-s + 0.400·40-s − 0.822·41-s − 0.0408·43-s + 0.764·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.732T + 2T^{2} \) |
| 7 | \( 1 + 4.46T + 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 17 | \( 1 - 6.73T + 17T^{2} \) |
| 19 | \( 1 - 5.46T + 19T^{2} \) |
| 23 | \( 1 - 0.535T + 23T^{2} \) |
| 29 | \( 1 - 2.73T + 29T^{2} \) |
| 31 | \( 1 - 3.19T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 5.26T + 41T^{2} \) |
| 43 | \( 1 + 0.267T + 43T^{2} \) |
| 47 | \( 1 + 0.196T + 47T^{2} \) |
| 53 | \( 1 - 6.92T + 53T^{2} \) |
| 59 | \( 1 + 7.26T + 59T^{2} \) |
| 61 | \( 1 - 4.46T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 - 15.3T + 73T^{2} \) |
| 79 | \( 1 - 1.92T + 79T^{2} \) |
| 83 | \( 1 + 2.53T + 83T^{2} \) |
| 89 | \( 1 + 1.26T + 89T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
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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
−7.54148415053992095658560073741, −6.79049618465863224590740014512, −5.93071805067082377411300814757, −5.42790696920699000623971358107, −4.76864451751183392411411144937, −3.73013629301501020589259344855, −3.23086089839306556243710949580, −2.79488583044377622843072655319, −0.966497683811036615818317136827, 0,
0.966497683811036615818317136827, 2.79488583044377622843072655319, 3.23086089839306556243710949580, 3.73013629301501020589259344855, 4.76864451751183392411411144937, 5.42790696920699000623971358107, 5.93071805067082377411300814757, 6.79049618465863224590740014512, 7.54148415053992095658560073741
