| L(s) = 1 | + 78·2-s + 243·3-s + 4.03e3·4-s + 5.37e3·5-s + 1.89e4·6-s + 1.55e5·8-s + 5.90e4·9-s + 4.18e5·10-s + 6.37e5·11-s + 9.80e5·12-s − 7.66e5·13-s + 1.30e6·15-s + 3.82e6·16-s − 3.08e6·17-s + 4.60e6·18-s + 1.95e7·19-s + 2.16e7·20-s + 4.97e7·22-s + 1.53e7·23-s + 3.76e7·24-s − 1.99e7·25-s − 5.97e7·26-s + 1.43e7·27-s + 1.07e7·29-s + 1.01e8·30-s + 5.09e7·31-s − 1.88e7·32-s + ⋯ |
| L(s) = 1 | + 1.72·2-s + 0.577·3-s + 1.97·4-s + 0.768·5-s + 0.995·6-s + 1.67·8-s + 1/3·9-s + 1.32·10-s + 1.19·11-s + 1.13·12-s − 0.572·13-s + 0.443·15-s + 0.912·16-s − 0.526·17-s + 0.574·18-s + 1.80·19-s + 1.51·20-s + 2.05·22-s + 0.496·23-s + 0.965·24-s − 0.409·25-s − 0.986·26-s + 0.192·27-s + 0.0973·29-s + 0.764·30-s + 0.319·31-s − 0.0995·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(10.72979720\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.72979720\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p^{5} T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 39 p T + p^{11} T^{2} \) |
| 5 | \( 1 - 1074 p T + p^{11} T^{2} \) |
| 11 | \( 1 - 637836 T + p^{11} T^{2} \) |
| 13 | \( 1 + 766214 T + p^{11} T^{2} \) |
| 17 | \( 1 + 3084354 T + p^{11} T^{2} \) |
| 19 | \( 1 - 1026916 p T + p^{11} T^{2} \) |
| 23 | \( 1 - 15312360 T + p^{11} T^{2} \) |
| 29 | \( 1 - 10751262 T + p^{11} T^{2} \) |
| 31 | \( 1 - 50937400 T + p^{11} T^{2} \) |
| 37 | \( 1 - 664740830 T + p^{11} T^{2} \) |
| 41 | \( 1 + 898833450 T + p^{11} T^{2} \) |
| 43 | \( 1 + 957947188 T + p^{11} T^{2} \) |
| 47 | \( 1 - 1555741344 T + p^{11} T^{2} \) |
| 53 | \( 1 - 3792417030 T + p^{11} T^{2} \) |
| 59 | \( 1 + 555306924 T + p^{11} T^{2} \) |
| 61 | \( 1 + 4950420998 T + p^{11} T^{2} \) |
| 67 | \( 1 - 5292399284 T + p^{11} T^{2} \) |
| 71 | \( 1 + 14831086248 T + p^{11} T^{2} \) |
| 73 | \( 1 + 13971005210 T + p^{11} T^{2} \) |
| 79 | \( 1 - 3720542360 T + p^{11} T^{2} \) |
| 83 | \( 1 + 8768454036 T + p^{11} T^{2} \) |
| 89 | \( 1 - 25472769174 T + p^{11} T^{2} \) |
| 97 | \( 1 - 39092494846 T + p^{11} T^{2} \) |
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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
−11.48749377830512529781957505970, −9.991780732873669881337443233294, −9.072162522226837247428790710748, −7.40293034485645209112067913698, −6.48574741464953996249788239574, −5.46991660137401294792930132621, −4.44104828664399765491545646750, −3.37660372302341633166789549210, −2.42446180839470704635213245907, −1.32371102469998342998221924965,
1.32371102469998342998221924965, 2.42446180839470704635213245907, 3.37660372302341633166789549210, 4.44104828664399765491545646750, 5.46991660137401294792930132621, 6.48574741464953996249788239574, 7.40293034485645209112067913698, 9.072162522226837247428790710748, 9.991780732873669881337443233294, 11.48749377830512529781957505970
