| L(s) = 1 | + 4·4-s + 12·16-s + 14·19-s − 8·31-s + 13·49-s − 2·61-s + 32·64-s + 56·76-s − 34·79-s − 4·109-s − 22·121-s − 32·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 2·4-s + 3·16-s + 3.21·19-s − 1.43·31-s + 13/7·49-s − 0.256·61-s + 4·64-s + 6.42·76-s − 3.82·79-s − 0.383·109-s − 2·121-s − 2.87·124-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 + 1/13·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.745214968\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.745214968\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 47 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 109 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 17 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 167 T^{2} + p^{2} T^{4} \) |
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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
−10.72335582525651327286945243932, −10.37869992966320522040361636092, −9.892953678818748897087035833162, −9.630718206181945748858291931371, −8.984480149682334454217962379159, −8.561540539951442997616302956568, −7.71414239063728191364340650517, −7.61682816812887264017793417737, −7.09568197028866990652976151612, −7.08584419907408387190388039722, −6.15016421990014445663002686297, −5.91941089447117372724921654593, −5.27331407988099272678167746921, −5.17105547596366507830763982795, −3.90199364691910903837224317661, −3.58879831536249008809439271590, −2.74074462213609618402113221827, −2.71621568371584767415814027516, −1.58581217443923666331756503862, −1.18007037306050144969940917793,
1.18007037306050144969940917793, 1.58581217443923666331756503862, 2.71621568371584767415814027516, 2.74074462213609618402113221827, 3.58879831536249008809439271590, 3.90199364691910903837224317661, 5.17105547596366507830763982795, 5.27331407988099272678167746921, 5.91941089447117372724921654593, 6.15016421990014445663002686297, 7.08584419907408387190388039722, 7.09568197028866990652976151612, 7.61682816812887264017793417737, 7.71414239063728191364340650517, 8.561540539951442997616302956568, 8.984480149682334454217962379159, 9.630718206181945748858291931371, 9.892953678818748897087035833162, 10.37869992966320522040361636092, 10.72335582525651327286945243932
