| L(s) = 1 | + 2.18·2-s + 2.79·4-s + 1.73·8-s − 6.10·11-s − 1.79·16-s − 13.3·22-s − 5.29·23-s − 5·25-s + 10.5·29-s − 7.38·32-s − 12.1·37-s − 10.5·43-s − 17.0·44-s − 11.5·46-s − 10.9·50-s + 3.36·53-s + 23.1·58-s − 12.5·64-s + 11.7·67-s + 11.2·71-s − 26.6·74-s − 17.7·79-s − 23.1·86-s − 10.5·88-s − 14.7·92-s − 13.9·100-s + 7.37·106-s + ⋯ |
| L(s) = 1 | + 1.54·2-s + 1.39·4-s + 0.612·8-s − 1.84·11-s − 0.447·16-s − 2.85·22-s − 1.10·23-s − 25-s + 1.96·29-s − 1.30·32-s − 1.99·37-s − 1.61·43-s − 2.57·44-s − 1.70·46-s − 1.54·50-s + 0.462·53-s + 3.04·58-s − 1.57·64-s + 1.43·67-s + 1.33·71-s − 3.09·74-s − 1.99·79-s − 2.49·86-s − 1.12·88-s − 1.53·92-s − 1.39·100-s + 0.716·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.18T + 2T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 6.10T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 5.29T + 23T^{2} \) |
| 29 | \( 1 - 10.5T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 12.1T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 3.36T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 17.7T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−8.076678727051654260283869948567, −7.08455410905175765751120533634, −6.43912793176248262589905604059, −5.52336271523020168941915072703, −5.16762747091273900619611548079, −4.36126038913861218379494643212, −3.49673628288320842682282396417, −2.73454580224607483850903244392, −1.96418117660714103075836856817, 0,
1.96418117660714103075836856817, 2.73454580224607483850903244392, 3.49673628288320842682282396417, 4.36126038913861218379494643212, 5.16762747091273900619611548079, 5.52336271523020168941915072703, 6.43912793176248262589905604059, 7.08455410905175765751120533634, 8.076678727051654260283869948567
