| L(s) = 1 | + 8.19e3·2-s + 1.96e5·3-s + 6.71e7·4-s − 1.22e9·5-s + 1.61e9·6-s − 1.13e11·7-s + 5.49e11·8-s − 2.50e12·9-s − 1.00e13·10-s + 1.31e13·12-s − 9.29e14·14-s − 2.39e14·15-s + 4.50e15·16-s − 2.05e16·18-s − 8.19e16·20-s − 2.22e16·21-s + 7.67e17·23-s + 1.08e17·24-s + 1.49e18·25-s − 9.91e17·27-s − 7.61e18·28-s + 1.93e19·29-s − 1.96e18·30-s + 3.68e19·32-s + 1.38e20·35-s − 1.67e20·36-s − 6.71e20·40-s + ⋯ |
| L(s) = 1 | + 2-s + 0.123·3-s + 4-s − 5-s + 0.123·6-s − 1.17·7-s + 8-s − 0.984·9-s − 10-s + 0.123·12-s − 1.17·14-s − 0.123·15-s + 16-s − 0.984·18-s − 20-s − 0.144·21-s + 1.52·23-s + 0.123·24-s + 25-s − 0.244·27-s − 1.17·28-s + 1.88·29-s − 0.123·30-s + 32-s + 1.17·35-s − 0.984·36-s − 40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(27-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+13) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{27}{2})\) |
\(\approx\) |
\(2.883230092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.883230092\) |
| \(L(14)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{13} T \) |
| 5 | \( 1 + p^{13} T \) |
| good | 3 | \( 1 - 196556 T + p^{26} T^{2} \) |
| 7 | \( 1 + 113407192076 T + p^{26} T^{2} \) |
| 11 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 13 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 17 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 19 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 23 | \( 1 - 767155019694663556 T + p^{26} T^{2} \) |
| 29 | \( 1 - 19310164803613363658 T + p^{26} T^{2} \) |
| 31 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 37 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 41 | \( 1 + \)\(18\!\cdots\!38\)\( T + p^{26} T^{2} \) |
| 43 | \( 1 - \)\(34\!\cdots\!96\)\( T + p^{26} T^{2} \) |
| 47 | \( 1 - \)\(57\!\cdots\!44\)\( T + p^{26} T^{2} \) |
| 53 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 59 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 61 | \( 1 + \)\(50\!\cdots\!58\)\( T + p^{26} T^{2} \) |
| 67 | \( 1 + \)\(94\!\cdots\!76\)\( T + p^{26} T^{2} \) |
| 71 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 73 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 79 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 83 | \( 1 + \)\(17\!\cdots\!84\)\( T + p^{26} T^{2} \) |
| 89 | \( 1 - \)\(23\!\cdots\!58\)\( T + p^{26} T^{2} \) |
| 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
−12.66549942536772781440206254687, −11.74067586160998547596472729429, −10.55532799038991259664972302010, −8.736120931456746373496889850548, −7.27078758750976837871519238486, −6.18639787514492437323572207705, −4.76071837132538784652376469713, −3.40411104409083400643037773205, −2.76661905072831203458327612173, −0.70737818229352044163842818386,
0.70737818229352044163842818386, 2.76661905072831203458327612173, 3.40411104409083400643037773205, 4.76071837132538784652376469713, 6.18639787514492437323572207705, 7.27078758750976837871519238486, 8.736120931456746373496889850548, 10.55532799038991259664972302010, 11.74067586160998547596472729429, 12.66549942536772781440206254687
