| L(s) = 1 | + 1.14·2-s − 3-s − 0.697·4-s + 0.866·5-s − 1.14·6-s − 4.61·7-s − 3.07·8-s + 9-s + 0.988·10-s − 2.50·11-s + 0.697·12-s − 0.0237·13-s − 5.27·14-s − 0.866·15-s − 2.11·16-s + 2.24·17-s + 1.14·18-s − 6.29·19-s − 0.604·20-s + 4.61·21-s − 2.85·22-s + 0.404·23-s + 3.07·24-s − 4.24·25-s − 0.0270·26-s − 27-s + 3.22·28-s + ⋯ |
| L(s) = 1 | + 0.806·2-s − 0.577·3-s − 0.348·4-s + 0.387·5-s − 0.465·6-s − 1.74·7-s − 1.08·8-s + 0.333·9-s + 0.312·10-s − 0.755·11-s + 0.201·12-s − 0.00658·13-s − 1.40·14-s − 0.223·15-s − 0.529·16-s + 0.544·17-s + 0.268·18-s − 1.44·19-s − 0.135·20-s + 1.00·21-s − 0.609·22-s + 0.0843·23-s + 0.628·24-s − 0.849·25-s − 0.00531·26-s − 0.192·27-s + 0.609·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 - 1.14T + 2T^{2} \) |
| 5 | \( 1 - 0.866T + 5T^{2} \) |
| 7 | \( 1 + 4.61T + 7T^{2} \) |
| 11 | \( 1 + 2.50T + 11T^{2} \) |
| 13 | \( 1 + 0.0237T + 13T^{2} \) |
| 17 | \( 1 - 2.24T + 17T^{2} \) |
| 19 | \( 1 + 6.29T + 19T^{2} \) |
| 23 | \( 1 - 0.404T + 23T^{2} \) |
| 29 | \( 1 - 2.03T + 29T^{2} \) |
| 31 | \( 1 - 2.82T + 31T^{2} \) |
| 37 | \( 1 - 4.22T + 37T^{2} \) |
| 41 | \( 1 + 1.78T + 41T^{2} \) |
| 43 | \( 1 - 3.67T + 43T^{2} \) |
| 47 | \( 1 + 7.27T + 47T^{2} \) |
| 53 | \( 1 + 11.9T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 + 5.99T + 61T^{2} \) |
| 67 | \( 1 - 9.33T + 67T^{2} \) |
| 71 | \( 1 - 7.22T + 71T^{2} \) |
| 73 | \( 1 - 17.0T + 73T^{2} \) |
| 79 | \( 1 + 9.72T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 + 7.63T + 97T^{2} \) |
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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
−10.89174375934908685016530689891, −9.902673452829412381824643643161, −9.399155926713699149530353626016, −8.070361571606850277042981625536, −6.49327716353203182203794074343, −6.10225582149541121995427059033, −5.03358651955702708694604375480, −3.86922863066290409006924865744, −2.75985771699596474722601184236, 0,
2.75985771699596474722601184236, 3.86922863066290409006924865744, 5.03358651955702708694604375480, 6.10225582149541121995427059033, 6.49327716353203182203794074343, 8.070361571606850277042981625536, 9.399155926713699149530353626016, 9.902673452829412381824643643161, 10.89174375934908685016530689891
