| L(s) = 1 | + 456·2-s + 5.06e4·3-s − 3.16e5·4-s − 2.37e6·5-s + 2.30e7·6-s − 1.69e7·7-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 − 7.71e9·14-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 − 8.56e11·21-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 + 5.35e12·28-s + ⋯ |
| L(s) = 1 | + 0.629·2-s + 1.48·3-s − 0.603·4-s − 0.544·5-s + 0.935·6-s − 0.158·7-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 − 0.0997·14-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 − 0.235·21-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 + 0.0956·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(20-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+19/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(10)\) |
\(\approx\) |
\(1.981735405\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.981735405\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 57 p^{3} T + p^{19} T^{2} \) |
| 3 | \( 1 - 1876 p^{3} T + p^{19} T^{2} \) |
| 5 | \( 1 + 475482 p T + p^{19} T^{2} \) |
| 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
−30.55672605125458804006448947936, −27.29445208791780927881588765643, −25.59118621334058836769515346551, −23.37831105449752633139466303810, −21.00993031068581648326307127139, −19.01545399887427581559957963587, −15.00403257750766652703892999605, −13.31064190731384332629695179807, −8.677106764116385551617153434409, −3.60719823716604384253001609923,
3.60719823716604384253001609923, 8.677106764116385551617153434409, 13.31064190731384332629695179807, 15.00403257750766652703892999605, 19.01545399887427581559957963587, 21.00993031068581648326307127139, 23.37831105449752633139466303810, 25.59118621334058836769515346551, 27.29445208791780927881588765643, 30.55672605125458804006448947936
