| L(s) = 1 | + 1.12·2-s − 1.52·3-s − 0.730·4-s − 5-s − 1.71·6-s − 3.07·8-s − 0.680·9-s − 1.12·10-s + 11-s + 1.11·12-s + 6.04·13-s + 1.52·15-s − 2.00·16-s + 1.11·17-s − 0.766·18-s + 4.14·19-s + 0.730·20-s + 1.12·22-s − 1.82·23-s + 4.68·24-s + 25-s + 6.81·26-s + 5.60·27-s + 1.03·29-s + 1.71·30-s − 10.3·31-s + 3.89·32-s + ⋯ |
| L(s) = 1 | + 0.796·2-s − 0.879·3-s − 0.365·4-s − 0.447·5-s − 0.700·6-s − 1.08·8-s − 0.226·9-s − 0.356·10-s + 0.301·11-s + 0.321·12-s + 1.67·13-s + 0.393·15-s − 0.501·16-s + 0.269·17-s − 0.180·18-s + 0.950·19-s + 0.163·20-s + 0.240·22-s − 0.381·23-s + 0.956·24-s + 0.200·25-s + 1.33·26-s + 1.07·27-s + 0.191·29-s + 0.313·30-s − 1.85·31-s + 0.688·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 1.12T + 2T^{2} \) |
| 3 | \( 1 + 1.52T + 3T^{2} \) |
| 13 | \( 1 - 6.04T + 13T^{2} \) |
| 17 | \( 1 - 1.11T + 17T^{2} \) |
| 19 | \( 1 - 4.14T + 19T^{2} \) |
| 23 | \( 1 + 1.82T + 23T^{2} \) |
| 29 | \( 1 - 1.03T + 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 + 7.41T + 37T^{2} \) |
| 41 | \( 1 - 4.44T + 41T^{2} \) |
| 43 | \( 1 + 5.56T + 43T^{2} \) |
| 47 | \( 1 - 4.50T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 - 7.44T + 59T^{2} \) |
| 61 | \( 1 + 6.45T + 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 - 1.84T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 - 1.50T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 + 14.0T + 89T^{2} \) |
| 97 | \( 1 - 5.85T + 97T^{2} \) |
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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
−8.647498277952993671405047241189, −7.61036169885011168205506253674, −6.62699559547224357001007392656, −5.84512695243387101542212818237, −5.48091822966453944900911769556, −4.54984257082971860593027292311, −3.69071541470489063737774498424, −3.13869892892609427259392816037, −1.32381847342350111962033295941, 0,
1.32381847342350111962033295941, 3.13869892892609427259392816037, 3.69071541470489063737774498424, 4.54984257082971860593027292311, 5.48091822966453944900911769556, 5.84512695243387101542212818237, 6.62699559547224357001007392656, 7.61036169885011168205506253674, 8.647498277952993671405047241189
