| L(s) = 1 | − 2.48·2-s + 0.674·3-s + 4.18·4-s − 1.47·5-s − 1.67·6-s + 0.0650·7-s − 5.43·8-s − 2.54·9-s + 3.67·10-s + 3.92·11-s + 2.82·12-s + 5.49·13-s − 0.161·14-s − 0.996·15-s + 5.15·16-s − 3.93·17-s + 6.32·18-s + 4.25·19-s − 6.17·20-s + 0.0439·21-s − 9.75·22-s − 9.41·23-s − 3.66·24-s − 2.82·25-s − 13.6·26-s − 3.74·27-s + 0.272·28-s + ⋯ |
| L(s) = 1 | − 1.75·2-s + 0.389·3-s + 2.09·4-s − 0.659·5-s − 0.685·6-s + 0.0246·7-s − 1.92·8-s − 0.848·9-s + 1.16·10-s + 1.18·11-s + 0.815·12-s + 1.52·13-s − 0.0432·14-s − 0.257·15-s + 1.28·16-s − 0.955·17-s + 1.49·18-s + 0.976·19-s − 1.38·20-s + 0.00958·21-s − 2.07·22-s − 1.96·23-s − 0.749·24-s − 0.564·25-s − 2.67·26-s − 0.720·27-s + 0.0514·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2671 | \( 1 + T \) |
| good | 2 | \( 1 + 2.48T + 2T^{2} \) |
| 3 | \( 1 - 0.674T + 3T^{2} \) |
| 5 | \( 1 + 1.47T + 5T^{2} \) |
| 7 | \( 1 - 0.0650T + 7T^{2} \) |
| 11 | \( 1 - 3.92T + 11T^{2} \) |
| 13 | \( 1 - 5.49T + 13T^{2} \) |
| 17 | \( 1 + 3.93T + 17T^{2} \) |
| 19 | \( 1 - 4.25T + 19T^{2} \) |
| 23 | \( 1 + 9.41T + 23T^{2} \) |
| 29 | \( 1 + 1.89T + 29T^{2} \) |
| 31 | \( 1 + 3.01T + 31T^{2} \) |
| 37 | \( 1 - 0.553T + 37T^{2} \) |
| 41 | \( 1 - 0.496T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 - 2.40T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 + 11.9T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 - 3.89T + 71T^{2} \) |
| 73 | \( 1 - 9.38T + 73T^{2} \) |
| 79 | \( 1 - 12.5T + 79T^{2} \) |
| 83 | \( 1 - 5.42T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 - 7.15T + 97T^{2} \) |
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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
−8.527867861146677478238660242844, −7.949122958265698901870927966527, −7.38067411800735301189585583088, −6.30475070707296633217058931062, −5.91451908159216207098221309057, −4.12832570567142106705994870587, −3.47581934021781223518867907552, −2.24997350085846292287204895413, −1.29737593331410231321617508566, 0,
1.29737593331410231321617508566, 2.24997350085846292287204895413, 3.47581934021781223518867907552, 4.12832570567142106705994870587, 5.91451908159216207098221309057, 6.30475070707296633217058931062, 7.38067411800735301189585583088, 7.949122958265698901870927966527, 8.527867861146677478238660242844
