| L(s) = 1 | − 9-s − 6·11-s + 8·19-s − 6·29-s − 20·31-s − 49-s − 24·59-s − 8·61-s + 18·71-s − 34·79-s + 81-s + 6·99-s + 12·101-s + 14·109-s + 5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s − 8·171-s + 173-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 1.80·11-s + 1.83·19-s − 1.11·29-s − 3.59·31-s − 1/7·49-s − 3.12·59-s − 1.02·61-s + 2.13·71-s − 3.82·79-s + 1/9·81-s + 0.603·99-s + 1.19·101-s + 1.34·109-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s − 0.611·171-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4401586343\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4401586343\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 17 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−9.557473680068734083830349075180, −8.970545750157528644032565398164, −8.650854741330197126309256332909, −7.81828475368748484503176337618, −7.78033566376799056782608340854, −7.37786403554171229777510510470, −7.31014788506635078761734333576, −6.54680089435997162472751484423, −5.99717163914688994145721380164, −5.52835759881352755659698140166, −5.48097599678030021515376073481, −4.98622126509834413679321234720, −4.60855972616640113046761949705, −3.70801032300711136872989797930, −3.59966988523185306072903141167, −2.93053430493917176604976833997, −2.65251916893691641455894072769, −1.82455179093352505784715913440, −1.47015112093180162887652636033, −0.22576658699438183689429287858,
0.22576658699438183689429287858, 1.47015112093180162887652636033, 1.82455179093352505784715913440, 2.65251916893691641455894072769, 2.93053430493917176604976833997, 3.59966988523185306072903141167, 3.70801032300711136872989797930, 4.60855972616640113046761949705, 4.98622126509834413679321234720, 5.48097599678030021515376073481, 5.52835759881352755659698140166, 5.99717163914688994145721380164, 6.54680089435997162472751484423, 7.31014788506635078761734333576, 7.37786403554171229777510510470, 7.78033566376799056782608340854, 7.81828475368748484503176337618, 8.650854741330197126309256332909, 8.970545750157528644032565398164, 9.557473680068734083830349075180
