| L(s) = 1 | + 8.14e3·2-s − 5.31e5·3-s + 4.95e7·4-s − 2.44e8·5-s − 4.32e9·6-s + 2.66e11·8-s + 2.82e11·9-s − 1.98e12·10-s − 2.63e13·12-s + 1.29e14·15-s + 1.34e15·16-s − 4.32e14·17-s + 2.29e15·18-s + 1.90e15·19-s − 1.20e16·20-s + 3.72e16·23-s − 1.41e17·24-s + 5.96e16·25-s − 1.50e17·27-s + 1.05e18·30-s + 1.45e18·31-s + 6.44e18·32-s − 3.52e18·34-s + 1.39e19·36-s + 1.55e19·38-s − 6.51e19·40-s − 6.89e19·45-s + ⋯ |
| L(s) = 1 | + 1.98·2-s − 3-s + 2.95·4-s − 5-s − 1.98·6-s + 3.88·8-s + 9-s − 1.98·10-s − 2.95·12-s + 15-s + 4.76·16-s − 0.742·17-s + 1.98·18-s + 0.860·19-s − 2.95·20-s + 1.69·23-s − 3.88·24-s + 25-s − 27-s + 1.98·30-s + 1.85·31-s + 5.58·32-s − 1.47·34-s + 2.95·36-s + 1.71·38-s − 3.88·40-s − 45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(25-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+12) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{25}{2})\) |
\(\approx\) |
\(5.527767141\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.527767141\) |
| \(L(13)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p^{12} T \) |
| 5 | \( 1 + p^{12} T \) |
| good | 2 | \( 1 - 8143 T + p^{24} T^{2} \) |
| 7 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 11 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 13 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 17 | \( 1 + 432524144062082 T + p^{24} T^{2} \) |
| 19 | \( 1 - 1904424373265762 T + p^{24} T^{2} \) |
| 23 | \( 1 - 37214344554868798 T + p^{24} T^{2} \) |
| 29 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 31 | \( 1 - 1458725602323329282 T + p^{24} T^{2} \) |
| 37 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 41 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 43 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 47 | \( 1 - 51418335280297668478 T + p^{24} T^{2} \) |
| 53 | \( 1 + \)\(34\!\cdots\!22\)\( T + p^{24} T^{2} \) |
| 59 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 61 | \( 1 + \)\(52\!\cdots\!18\)\( T + p^{24} T^{2} \) |
| 67 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 71 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 73 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 79 | \( 1 + \)\(20\!\cdots\!78\)\( T + p^{24} T^{2} \) |
| 83 | \( 1 - \)\(25\!\cdots\!38\)\( T + p^{24} T^{2} \) |
| 89 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 97 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
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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
−13.55571330915529829283217657340, −12.41604745351859331430990182719, −11.58197295983240027769016140130, −10.72314002369396620659421421191, −7.44411685358830076027684109270, −6.47556373397802188895733843288, −5.08693055766113894876843730158, −4.27519131185009954580365468712, −2.94980983629975211166127980006, −1.08793500955262502783059938238,
1.08793500955262502783059938238, 2.94980983629975211166127980006, 4.27519131185009954580365468712, 5.08693055766113894876843730158, 6.47556373397802188895733843288, 7.44411685358830076027684109270, 10.72314002369396620659421421191, 11.58197295983240027769016140130, 12.41604745351859331430990182719, 13.55571330915529829283217657340
