| L(s) = 1 | + 0.692·2-s + 3-s − 1.52·4-s − 2.78·5-s + 0.692·6-s + 7-s − 2.43·8-s + 9-s − 1.92·10-s − 0.348·11-s − 1.52·12-s + 2.17·13-s + 0.692·14-s − 2.78·15-s + 1.35·16-s + 7.42·17-s + 0.692·18-s − 0.255·19-s + 4.22·20-s + 21-s − 0.241·22-s − 4.98·23-s − 2.43·24-s + 2.73·25-s + 1.50·26-s + 27-s − 1.52·28-s + ⋯ |
| L(s) = 1 | + 0.489·2-s + 0.577·3-s − 0.760·4-s − 1.24·5-s + 0.282·6-s + 0.377·7-s − 0.861·8-s + 0.333·9-s − 0.608·10-s − 0.105·11-s − 0.438·12-s + 0.603·13-s + 0.185·14-s − 0.718·15-s + 0.338·16-s + 1.80·17-s + 0.163·18-s − 0.0586·19-s + 0.945·20-s + 0.218·21-s − 0.0514·22-s − 1.03·23-s − 0.497·24-s + 0.547·25-s + 0.295·26-s + 0.192·27-s − 0.287·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 - 0.692T + 2T^{2} \) |
| 5 | \( 1 + 2.78T + 5T^{2} \) |
| 11 | \( 1 + 0.348T + 11T^{2} \) |
| 13 | \( 1 - 2.17T + 13T^{2} \) |
| 17 | \( 1 - 7.42T + 17T^{2} \) |
| 19 | \( 1 + 0.255T + 19T^{2} \) |
| 23 | \( 1 + 4.98T + 23T^{2} \) |
| 29 | \( 1 + 6.88T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 + 1.64T + 37T^{2} \) |
| 41 | \( 1 - 3.56T + 41T^{2} \) |
| 43 | \( 1 + 9.34T + 43T^{2} \) |
| 47 | \( 1 - 9.73T + 47T^{2} \) |
| 53 | \( 1 + 7.44T + 53T^{2} \) |
| 59 | \( 1 + 2.01T + 59T^{2} \) |
| 61 | \( 1 + 7.69T + 61T^{2} \) |
| 67 | \( 1 + 6.52T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 + 9.37T + 73T^{2} \) |
| 79 | \( 1 + 5.92T + 79T^{2} \) |
| 83 | \( 1 - 1.68T + 83T^{2} \) |
| 89 | \( 1 + 17.6T + 89T^{2} \) |
| 97 | \( 1 + 0.216T + 97T^{2} \) |
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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
−8.295926829838730603213955931291, −7.86250425310618428354266566388, −7.25001629553438894140287985788, −5.87328138432475504780655383797, −5.30119380499332497139575449503, −4.18486869708857855341031591002, −3.76081292997925957555670404027, −3.11133182178127022360390571296, −1.51945173405302223573002331817, 0,
1.51945173405302223573002331817, 3.11133182178127022360390571296, 3.76081292997925957555670404027, 4.18486869708857855341031591002, 5.30119380499332497139575449503, 5.87328138432475504780655383797, 7.25001629553438894140287985788, 7.86250425310618428354266566388, 8.295926829838730603213955931291
