| L(s) = 1 | − 2·4-s − 3·7-s + 6·11-s + 7·13-s + 6·19-s − 6·23-s − 2·25-s + 6·28-s + 6·29-s + 12·41-s + 43-s − 12·44-s − 49-s − 14·52-s − 24·53-s + 6·59-s − 61-s + 8·64-s + 15·67-s + 18·71-s − 12·76-s − 18·77-s − 22·79-s − 12·89-s − 21·91-s + 12·92-s − 9·97-s + ⋯ |
| L(s) = 1 | − 4-s − 1.13·7-s + 1.80·11-s + 1.94·13-s + 1.37·19-s − 1.25·23-s − 2/5·25-s + 1.13·28-s + 1.11·29-s + 1.87·41-s + 0.152·43-s − 1.80·44-s − 1/7·49-s − 1.94·52-s − 3.29·53-s + 0.781·59-s − 0.128·61-s + 64-s + 1.83·67-s + 2.13·71-s − 1.37·76-s − 2.05·77-s − 2.47·79-s − 1.27·89-s − 2.20·91-s + 1.25·92-s − 0.913·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8782256358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8782256358\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 7 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 12 T + 89 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 6 T + 71 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 15 T + 142 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 18 T + 179 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 12 T + 137 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.04841022942684093750473571811, −13.30646623313535601246384551851, −12.76560893809697189766208200774, −12.45543798647023413397316385650, −11.62132858519400011466854118130, −11.35842360822776695384530967690, −10.71461154115753083249230894512, −9.702645435345651152911373397047, −9.539018328055378026618697972994, −9.294476075060665362368958814769, −8.330447422381186126035980313518, −8.215411476473491959655593083959, −7.06727364404547877490573285254, −6.30168178556469811099091670290, −6.22918150692400218279346685722, −5.28059496278388864830516026578, −4.09655384127961639852717399175, −3.95264982477068541653487925411, −3.09036504556615477107757497699, −1.24814228878813644848669456513,
1.24814228878813644848669456513, 3.09036504556615477107757497699, 3.95264982477068541653487925411, 4.09655384127961639852717399175, 5.28059496278388864830516026578, 6.22918150692400218279346685722, 6.30168178556469811099091670290, 7.06727364404547877490573285254, 8.215411476473491959655593083959, 8.330447422381186126035980313518, 9.294476075060665362368958814769, 9.539018328055378026618697972994, 9.702645435345651152911373397047, 10.71461154115753083249230894512, 11.35842360822776695384530967690, 11.62132858519400011466854118130, 12.45543798647023413397316385650, 12.76560893809697189766208200774, 13.30646623313535601246384551851, 14.04841022942684093750473571811
