| L(s) = 1 | − 456·2-s − 3.16e5·4-s + 2.37e6·5-s − 1.69e7·7-s + 3.83e8·8-s − 1.08e9·10-s + 1.62e7·11-s + 5.04e10·13-s + 7.71e9·14-s − 8.93e9·16-s − 2.25e11·17-s − 1.71e12·19-s − 7.52e11·20-s − 7.39e9·22-s − 1.40e13·23-s − 1.34e13·25-s − 2.29e13·26-s + 5.35e12·28-s − 1.13e12·29-s − 1.04e14·31-s − 1.96e14·32-s + 1.02e14·34-s − 4.02e13·35-s − 1.69e14·37-s + 7.79e14·38-s + 9.11e14·40-s + 3.30e15·41-s + ⋯ |
| L(s) = 1 | − 0.629·2-s − 0.603·4-s + 0.544·5-s − 0.158·7-s + 1.00·8-s − 0.342·10-s + 0.00207·11-s + 1.31·13-s + 0.0997·14-s − 0.0325·16-s − 0.460·17-s − 1.21·19-s − 0.328·20-s − 0.00130·22-s − 1.62·23-s − 0.703·25-s − 0.830·26-s + 0.0956·28-s − 0.0145·29-s − 0.710·31-s − 0.989·32-s + 0.289·34-s − 0.0862·35-s − 0.214·37-s + 0.765·38-s + 0.549·40-s + 1.57·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| good | 2 | \( 1 + 57 p^{3} T + p^{19} T^{2} \) |
| 5 | \( 1 - 475482 p T + p^{19} T^{2} \) |
| 7 | \( 1 + 345256 p^{2} T + p^{19} T^{2} \) |
| 11 | \( 1 - 1473828 p T + p^{19} T^{2} \) |
| 13 | \( 1 - 3878585774 p T + p^{19} T^{2} \) |
| 17 | \( 1 + 13239417618 p T + p^{19} T^{2} \) |
| 19 | \( 1 + 1710278572660 T + p^{19} T^{2} \) |
| 23 | \( 1 + 14036534788872 T + p^{19} T^{2} \) |
| 29 | \( 1 + 1137835269510 T + p^{19} T^{2} \) |
| 31 | \( 1 + 104626880141728 T + p^{19} T^{2} \) |
| 37 | \( 1 + 169392327370594 T + p^{19} T^{2} \) |
| 41 | \( 1 - 3309984750560838 T + p^{19} T^{2} \) |
| 43 | \( 1 - 1127913532193492 T + p^{19} T^{2} \) |
| 47 | \( 1 + 3498693987674256 T + p^{19} T^{2} \) |
| 53 | \( 1 + 29956294112980302 T + p^{19} T^{2} \) |
| 59 | \( 1 + 58391397642732420 T + p^{19} T^{2} \) |
| 61 | \( 1 - 23373685132672742 T + p^{19} T^{2} \) |
| 67 | \( 1 + 205102524257382244 T + p^{19} T^{2} \) |
| 71 | \( 1 - 177902341950417768 T + p^{19} T^{2} \) |
| 73 | \( 1 - 299853775038660122 T + p^{19} T^{2} \) |
| 79 | \( 1 + 92227090144007440 T + p^{19} T^{2} \) |
| 83 | \( 1 + 1208542823470585932 T + p^{19} T^{2} \) |
| 89 | \( 1 + 4371201192290304330 T + p^{19} T^{2} \) |
| 97 | \( 1 + 635013222218448094 T + p^{19} T^{2} \) |
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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
−16.04739768922484444483668698119, −14.10074091496582982562218717748, −12.97474252731123024576265644217, −10.79743574544804982641337848104, −9.439753376842828147554869134081, −8.172905330183021360295726408988, −6.06842387795516792449446418651, −4.09516699712993796838853653258, −1.71567017470694870363590472201, 0,
1.71567017470694870363590472201, 4.09516699712993796838853653258, 6.06842387795516792449446418651, 8.172905330183021360295726408988, 9.439753376842828147554869134081, 10.79743574544804982641337848104, 12.97474252731123024576265644217, 14.10074091496582982562218717748, 16.04739768922484444483668698119
