| L(s) = 1 | − 5.17·2-s − 3·3-s + 18.7·4-s + 15.5·6-s + 11.1·7-s − 55.5·8-s + 9·9-s − 11·11-s − 56.2·12-s + 89.5·13-s − 57.7·14-s + 137.·16-s − 58.3·17-s − 46.5·18-s + 24.5·19-s − 33.5·21-s + 56.8·22-s + 111.·23-s + 166.·24-s − 462.·26-s − 27·27-s + 209.·28-s + 109.·29-s + 119.·31-s − 265.·32-s + 33·33-s + 301.·34-s + ⋯ |
| L(s) = 1 | − 1.82·2-s − 0.577·3-s + 2.34·4-s + 1.05·6-s + 0.603·7-s − 2.45·8-s + 0.333·9-s − 0.301·11-s − 1.35·12-s + 1.91·13-s − 1.10·14-s + 2.14·16-s − 0.832·17-s − 0.609·18-s + 0.296·19-s − 0.348·21-s + 0.551·22-s + 1.01·23-s + 1.41·24-s − 3.49·26-s − 0.192·27-s + 1.41·28-s + 0.704·29-s + 0.692·31-s − 1.46·32-s + 0.174·33-s + 1.52·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8328825946\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8328825946\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 5.17T + 8T^{2} \) |
| 7 | \( 1 - 11.1T + 343T^{2} \) |
| 13 | \( 1 - 89.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 58.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 24.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 111.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 109.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 119.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 356.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 268.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 263.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 206.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 223.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 475.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 513.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 264.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.11e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 893.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.30e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.04e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.41e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 85.8T + 9.12e5T^{2} \) |
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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
−9.817313182668023106042341960427, −8.859440665637198509419379974986, −8.358633632858346343592121838122, −7.49636148418838884280929470073, −6.55752222548203851150900402441, −5.89254115554206835270875342554, −4.48202384216771394280160542389, −2.88293246248269445896109067128, −1.52498954813841366137012010758, −0.75058665251066263969345500238,
0.75058665251066263969345500238, 1.52498954813841366137012010758, 2.88293246248269445896109067128, 4.48202384216771394280160542389, 5.89254115554206835270875342554, 6.55752222548203851150900402441, 7.49636148418838884280929470073, 8.358633632858346343592121838122, 8.859440665637198509419379974986, 9.817313182668023106042341960427
