| L(s) = 1 | − 2-s − 3.17·3-s + 4-s − 2.18·5-s + 3.17·6-s + 3.13·7-s − 8-s + 7.05·9-s + 2.18·10-s − 3.17·12-s + 6.73·13-s − 3.13·14-s + 6.93·15-s + 16-s + 17-s − 7.05·18-s − 2.67·19-s − 2.18·20-s − 9.92·21-s − 9.30·23-s + 3.17·24-s − 0.216·25-s − 6.73·26-s − 12.8·27-s + 3.13·28-s + 0.265·29-s − 6.93·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.83·3-s + 0.5·4-s − 0.978·5-s + 1.29·6-s + 1.18·7-s − 0.353·8-s + 2.35·9-s + 0.691·10-s − 0.915·12-s + 1.86·13-s − 0.836·14-s + 1.79·15-s + 0.250·16-s + 0.242·17-s − 1.66·18-s − 0.613·19-s − 0.489·20-s − 2.16·21-s − 1.94·23-s + 0.647·24-s − 0.0432·25-s − 1.32·26-s − 2.47·27-s + 0.591·28-s + 0.0492·29-s − 1.26·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 3.17T + 3T^{2} \) |
| 5 | \( 1 + 2.18T + 5T^{2} \) |
| 7 | \( 1 - 3.13T + 7T^{2} \) |
| 13 | \( 1 - 6.73T + 13T^{2} \) |
| 19 | \( 1 + 2.67T + 19T^{2} \) |
| 23 | \( 1 + 9.30T + 23T^{2} \) |
| 29 | \( 1 - 0.265T + 29T^{2} \) |
| 31 | \( 1 + 4.70T + 31T^{2} \) |
| 37 | \( 1 + 2.39T + 37T^{2} \) |
| 41 | \( 1 - 4.58T + 41T^{2} \) |
| 43 | \( 1 - 5.82T + 43T^{2} \) |
| 47 | \( 1 + 4.44T + 47T^{2} \) |
| 53 | \( 1 - 2.88T + 53T^{2} \) |
| 59 | \( 1 + 3.36T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 + 7.10T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 - 4.81T + 79T^{2} \) |
| 83 | \( 1 - 8.36T + 83T^{2} \) |
| 89 | \( 1 - 1.26T + 89T^{2} \) |
| 97 | \( 1 + 3.38T + 97T^{2} \) |
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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
−8.034223220987598794885631787364, −7.41271674734273928503214039789, −6.49836876172777404356305970068, −5.92435454329829648700345393746, −5.27980403662179740989443896077, −4.16364872130703458462995374179, −3.87117025385688919010026610847, −1.88464113631243963328809214702, −1.08739922886565067750641066703, 0,
1.08739922886565067750641066703, 1.88464113631243963328809214702, 3.87117025385688919010026610847, 4.16364872130703458462995374179, 5.27980403662179740989443896077, 5.92435454329829648700345393746, 6.49836876172777404356305970068, 7.41271674734273928503214039789, 8.034223220987598794885631787364
