| L(s) = 1 | − 16·2-s − 6·3-s + 256·4-s + 560·5-s + 96·6-s − 2.40e3·7-s − 4.09e3·8-s − 1.96e4·9-s − 8.96e3·10-s − 5.41e4·11-s − 1.53e3·12-s − 1.13e5·13-s + 3.84e4·14-s − 3.36e3·15-s + 6.55e4·16-s + 6.26e3·17-s + 3.14e5·18-s + 2.57e5·19-s + 1.43e5·20-s + 1.44e4·21-s + 8.66e5·22-s − 2.66e5·23-s + 2.45e4·24-s − 1.63e6·25-s + 1.81e6·26-s + 2.35e5·27-s − 6.14e5·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.0427·3-s + 1/2·4-s + 0.400·5-s + 0.0302·6-s − 0.377·7-s − 0.353·8-s − 0.998·9-s − 0.283·10-s − 1.11·11-s − 0.0213·12-s − 1.09·13-s + 0.267·14-s − 0.0171·15-s + 1/4·16-s + 0.0181·17-s + 0.705·18-s + 0.452·19-s + 0.200·20-s + 0.0161·21-s + 0.788·22-s − 0.198·23-s + 0.0151·24-s − 0.839·25-s + 0.777·26-s + 0.0854·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{4} T \) |
| 7 | \( 1 + p^{4} T \) |
| good | 3 | \( 1 + 2 p T + p^{9} T^{2} \) |
| 5 | \( 1 - 112 p T + p^{9} T^{2} \) |
| 11 | \( 1 + 54152 T + p^{9} T^{2} \) |
| 13 | \( 1 + 113172 T + p^{9} T^{2} \) |
| 17 | \( 1 - 6262 T + p^{9} T^{2} \) |
| 19 | \( 1 - 257078 T + p^{9} T^{2} \) |
| 23 | \( 1 + 266000 T + p^{9} T^{2} \) |
| 29 | \( 1 - 1574714 T + p^{9} T^{2} \) |
| 31 | \( 1 + 4637484 T + p^{9} T^{2} \) |
| 37 | \( 1 + 11946238 T + p^{9} T^{2} \) |
| 41 | \( 1 - 21909126 T + p^{9} T^{2} \) |
| 43 | \( 1 - 27520592 T + p^{9} T^{2} \) |
| 47 | \( 1 - 52927836 T + p^{9} T^{2} \) |
| 53 | \( 1 - 16221222 T + p^{9} T^{2} \) |
| 59 | \( 1 + 140509618 T + p^{9} T^{2} \) |
| 61 | \( 1 + 202963560 T + p^{9} T^{2} \) |
| 67 | \( 1 - 153734572 T + p^{9} T^{2} \) |
| 71 | \( 1 - 3938816 p T + p^{9} T^{2} \) |
| 73 | \( 1 + 404022830 T + p^{9} T^{2} \) |
| 79 | \( 1 + 130689816 T + p^{9} T^{2} \) |
| 83 | \( 1 - 420134014 T + p^{9} T^{2} \) |
| 89 | \( 1 + 469542390 T + p^{9} T^{2} \) |
| 97 | \( 1 + 872501690 T + p^{9} T^{2} \) |
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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
−16.97156398772801953513680836714, −15.62762862669854132429460146263, −14.02868914399650773999320612569, −12.30375843851048355463008830546, −10.65802007288394348708776625087, −9.306427106328249338204083036979, −7.62305032881012570225622582573, −5.65468442777949473050849268788, −2.57769461149083632364105051475, 0,
2.57769461149083632364105051475, 5.65468442777949473050849268788, 7.62305032881012570225622582573, 9.306427106328249338204083036979, 10.65802007288394348708776625087, 12.30375843851048355463008830546, 14.02868914399650773999320612569, 15.62762862669854132429460146263, 16.97156398772801953513680836714
