| L(s) = 1 | + 4.15·2-s − 9·3-s − 14.7·4-s − 37.5·5-s − 37.3·6-s − 76.4·7-s − 194.·8-s + 81·9-s − 155.·10-s − 121·11-s + 132.·12-s + 169.·13-s − 317.·14-s + 337.·15-s − 333.·16-s − 0.875·17-s + 336.·18-s − 817.·19-s + 553.·20-s + 688.·21-s − 502.·22-s + 749.·23-s + 1.74e3·24-s − 1.71e3·25-s + 704.·26-s − 729·27-s + 1.12e3·28-s + ⋯ |
| L(s) = 1 | + 0.733·2-s − 0.577·3-s − 0.461·4-s − 0.671·5-s − 0.423·6-s − 0.589·7-s − 1.07·8-s + 0.333·9-s − 0.492·10-s − 0.301·11-s + 0.266·12-s + 0.278·13-s − 0.433·14-s + 0.387·15-s − 0.325·16-s − 0.000735·17-s + 0.244·18-s − 0.519·19-s + 0.309·20-s + 0.340·21-s − 0.221·22-s + 0.295·23-s + 0.619·24-s − 0.549·25-s + 0.204·26-s − 0.192·27-s + 0.272·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 - 4.15T + 32T^{2} \) |
| 5 | \( 1 + 37.5T + 3.12e3T^{2} \) |
| 7 | \( 1 + 76.4T + 1.68e4T^{2} \) |
| 13 | \( 1 - 169.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 0.875T + 1.41e6T^{2} \) |
| 19 | \( 1 + 817.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 749.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.04e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.47e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.58e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.62e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.82e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.28e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.17e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.16e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.40e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.69e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.75e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.04e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.21e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.19e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.46e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.38e5T + 8.58e9T^{2} \) |
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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
−15.13360148874937302477256187810, −13.69412194241532301953161295221, −12.66574831874663081872849108495, −11.68869704591268006977706955129, −10.11923600090473649603053583263, −8.491670798461644446160255407481, −6.58098047892767702440617356094, −5.05448953737723395622846367728, −3.58137809057447407668855622201, 0,
3.58137809057447407668855622201, 5.05448953737723395622846367728, 6.58098047892767702440617356094, 8.491670798461644446160255407481, 10.11923600090473649603053583263, 11.68869704591268006977706955129, 12.66574831874663081872849108495, 13.69412194241532301953161295221, 15.13360148874937302477256187810
