| L(s) = 1 | + 456·2-s − 5.06e4·3-s − 3.16e5·4-s + 2.37e6·5-s − 2.30e7·6-s − 3.83e8·8-s + 1.40e9·9-s + 1.08e9·10-s − 1.62e7·11-s + 1.60e10·12-s − 5.04e10·13-s − 1.20e11·15-s − 8.93e9·16-s − 2.25e11·17-s + 6.39e11·18-s + 1.71e12·19-s − 7.52e11·20-s − 7.39e9·22-s + 1.40e13·23-s + 1.94e13·24-s − 1.34e13·25-s − 2.29e13·26-s − 1.22e13·27-s + 1.13e12·29-s − 5.49e13·30-s + 1.04e14·31-s + 1.96e14·32-s + ⋯ |
| L(s) = 1 | + 0.629·2-s − 1.48·3-s − 0.603·4-s + 0.544·5-s − 0.935·6-s − 1.00·8-s + 1.20·9-s + 0.342·10-s − 0.00207·11-s + 0.896·12-s − 1.31·13-s − 0.808·15-s − 0.0325·16-s − 0.460·17-s + 0.760·18-s + 1.21·19-s − 0.328·20-s − 0.00130·22-s + 1.62·23-s + 1.50·24-s − 0.703·25-s − 0.830·26-s − 0.308·27-s + 0.0145·29-s − 0.509·30-s + 0.710·31-s + 0.989·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| good | 2 | \( 1 - 57 p^{3} T + p^{19} T^{2} \) |
| 3 | \( 1 + 1876 p^{3} T + p^{19} T^{2} \) |
| 5 | \( 1 - 475482 p T + p^{19} T^{2} \) |
| 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
−11.44848862108730209459593380489, −10.12893687601830269533975864871, −9.184023159054998373268946393633, −7.26317947198005261966997402888, −6.00754106158011764845484305360, −5.22014313993644252049188319506, −4.48467973948790478189621966521, −2.80366051864146297682676714893, −1.01371640477724463391822120969, 0,
1.01371640477724463391822120969, 2.80366051864146297682676714893, 4.48467973948790478189621966521, 5.22014313993644252049188319506, 6.00754106158011764845484305360, 7.26317947198005261966997402888, 9.184023159054998373268946393633, 10.12893687601830269533975864871, 11.44848862108730209459593380489
