| L(s) = 1 | − 2.33e3·2-s − 1.59e6·3-s − 6.16e7·4-s + 1.22e9·5-s + 3.72e9·6-s + 3.01e11·8-s + 2.54e12·9-s − 2.85e12·10-s + 9.82e13·12-s − 1.94e15·15-s + 3.43e15·16-s + 1.37e16·17-s − 5.94e15·18-s − 8.28e16·19-s − 7.52e16·20-s − 9.91e17·23-s − 4.80e17·24-s + 1.49e18·25-s − 4.05e18·27-s + 4.55e18·30-s + 1.98e19·31-s − 2.82e19·32-s − 3.21e19·34-s − 1.56e20·36-s + 1.93e20·38-s + 3.67e20·40-s + 3.10e21·45-s + ⋯ |
| L(s) = 1 | − 0.285·2-s − 3-s − 0.918·4-s + 5-s + 0.285·6-s + 0.547·8-s + 9-s − 0.285·10-s + 0.918·12-s − 15-s + 0.762·16-s + 1.38·17-s − 0.285·18-s − 1.97·19-s − 0.918·20-s − 1.96·23-s − 0.547·24-s + 25-s − 27-s + 0.285·30-s + 0.814·31-s − 0.765·32-s − 0.395·34-s − 0.918·36-s + 0.562·38-s + 0.547·40-s + 45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(27-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+13) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{27}{2})\) |
\(\approx\) |
\(1.010182764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.010182764\) |
| \(L(14)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p^{13} T \) |
| 5 | \( 1 - p^{13} T \) |
| good | 2 | \( 1 + 2339 T + p^{26} T^{2} \) |
| 7 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 11 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 13 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 17 | \( 1 - 13734556739205106 T + p^{26} T^{2} \) |
| 19 | \( 1 + 82854828439812022 T + p^{26} T^{2} \) |
| 23 | \( 1 + 991340034832255646 T + p^{26} T^{2} \) |
| 29 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 31 | \( 1 - 19887492532018026242 T + p^{26} T^{2} \) |
| 37 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 41 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 43 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 47 | \( 1 + \)\(10\!\cdots\!34\)\( T + p^{26} T^{2} \) |
| 53 | \( 1 - \)\(13\!\cdots\!34\)\( T + p^{26} T^{2} \) |
| 59 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 61 | \( 1 - \)\(31\!\cdots\!02\)\( T + p^{26} T^{2} \) |
| 67 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 71 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 73 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 79 | \( 1 - \)\(62\!\cdots\!38\)\( T + p^{26} T^{2} \) |
| 83 | \( 1 - \)\(45\!\cdots\!54\)\( T + p^{26} T^{2} \) |
| 89 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 97 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
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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
−13.30504390795227018457740177863, −12.25182702331133465945632220450, −10.43424497167971032878016815089, −9.779785551688119541594286383133, −8.196362719927737758337685714012, −6.35787016024571320806344683718, −5.34341656563995944278923323709, −4.09337478902474045167345278456, −1.84817330689088226033874469978, −0.60389580030631224682203098118,
0.60389580030631224682203098118, 1.84817330689088226033874469978, 4.09337478902474045167345278456, 5.34341656563995944278923323709, 6.35787016024571320806344683718, 8.196362719927737758337685714012, 9.779785551688119541594286383133, 10.43424497167971032878016815089, 12.25182702331133465945632220450, 13.30504390795227018457740177863
