| L(s) = 1 | + 512·2-s − 5.30e4·3-s + 2.62e5·4-s − 5.55e6·5-s − 2.71e7·6-s − 4.44e7·7-s + 1.34e8·8-s + 1.64e9·9-s − 2.84e9·10-s + 6.32e9·11-s − 1.39e10·12-s − 3.31e10·13-s − 2.27e10·14-s + 2.94e11·15-s + 6.87e10·16-s − 7.22e11·17-s + 8.44e11·18-s − 1.31e12·19-s − 1.45e12·20-s + 2.35e12·21-s + 3.23e12·22-s + 3.37e12·23-s − 7.11e12·24-s + 1.18e13·25-s − 1.69e13·26-s − 2.58e13·27-s − 1.16e13·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.55·3-s + 1/2·4-s − 1.27·5-s − 1.09·6-s − 0.416·7-s + 0.353·8-s + 1.41·9-s − 0.899·10-s + 0.808·11-s − 0.777·12-s − 0.866·13-s − 0.294·14-s + 1.97·15-s + 1/4·16-s − 1.47·17-s + 1.00·18-s − 0.933·19-s − 0.636·20-s + 0.648·21-s + 0.571·22-s + 0.391·23-s − 0.549·24-s + 0.618·25-s − 0.612·26-s − 0.652·27-s − 0.208·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{9} T \) |
| good | 3 | \( 1 + 1964 p^{3} T + p^{19} T^{2} \) |
| 5 | \( 1 + 1111386 p T + p^{19} T^{2} \) |
| 7 | \( 1 + 6356632 p T + p^{19} T^{2} \) |
| 11 | \( 1 - 574606812 p T + p^{19} T^{2} \) |
| 13 | \( 1 + 33124973098 T + p^{19} T^{2} \) |
| 17 | \( 1 + 42491485422 p T + p^{19} T^{2} \) |
| 19 | \( 1 + 1312620671860 T + p^{19} T^{2} \) |
| 23 | \( 1 - 3379752742152 T + p^{19} T^{2} \) |
| 29 | \( 1 + 29378097714810 T + p^{19} T^{2} \) |
| 31 | \( 1 - 131976476089952 T + p^{19} T^{2} \) |
| 37 | \( 1 + 466464103652194 T + p^{19} T^{2} \) |
| 41 | \( 1 - 1889447681239482 T + p^{19} T^{2} \) |
| 43 | \( 1 + 4323507451065388 T + p^{19} T^{2} \) |
| 47 | \( 1 - 12103384387771536 T + p^{19} T^{2} \) |
| 53 | \( 1 + 30593935900444338 T + p^{19} T^{2} \) |
| 59 | \( 1 - 9908742512283780 T + p^{19} T^{2} \) |
| 61 | \( 1 + 91638145794467098 T + p^{19} T^{2} \) |
| 67 | \( 1 + 103349440678278244 T + p^{19} T^{2} \) |
| 71 | \( 1 - 285448322456957592 T + p^{19} T^{2} \) |
| 73 | \( 1 - 875008267167254042 T + p^{19} T^{2} \) |
| 79 | \( 1 + 1081394522969090320 T + p^{19} T^{2} \) |
| 83 | \( 1 + 665085275193888948 T + p^{19} T^{2} \) |
| 89 | \( 1 + 2020985164277790390 T + p^{19} T^{2} \) |
| 97 | \( 1 + 12825578365118067934 T + p^{19} T^{2} \) |
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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
−23.05276239989389903411920004440, −22.11762280417523645318611837242, −19.52569344178777357525959063090, −17.03961370633913488581902506862, −15.49491446556203867298814658599, −12.38499480155881779294276405247, −11.18296215537182210750269782079, −6.67921857852025478297247090299, −4.43588104472406081847709277467, 0,
4.43588104472406081847709277467, 6.67921857852025478297247090299, 11.18296215537182210750269782079, 12.38499480155881779294276405247, 15.49491446556203867298814658599, 17.03961370633913488581902506862, 19.52569344178777357525959063090, 22.11762280417523645318611837242, 23.05276239989389903411920004440
