| L(s) = 1 | + 4-s + 3·5-s − 9-s − 3·16-s + 3·20-s + 4·25-s + 6·29-s − 36-s − 3·45-s − 10·49-s − 12·59-s − 7·64-s − 9·80-s − 8·81-s + 4·100-s + 10·109-s + 6·116-s + 17·121-s − 3·125-s + 127-s + 131-s + 137-s + 139-s + 3·144-s + 18·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1.34·5-s − 1/3·9-s − 3/4·16-s + 0.670·20-s + 4/5·25-s + 1.11·29-s − 1/6·36-s − 0.447·45-s − 1.42·49-s − 1.56·59-s − 7/8·64-s − 1.00·80-s − 8/9·81-s + 2/5·100-s + 0.957·109-s + 0.557·116-s + 1.54·121-s − 0.268·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/4·144-s + 1.49·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.528433200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.528433200\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| 29 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 41 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 89 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 113 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
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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
−10.82553789347932959955315112147, −10.28353392795342278987908972818, −9.737481940209185346237104800034, −9.339836646165358185420325475545, −8.718569936936276174746017611956, −8.197442728315647154964018015629, −7.41365440531891275049274903808, −6.78489189350551972813060491708, −6.24723087762616691306285278455, −5.86126892323299065677908591135, −5.03027375920784564217785266273, −4.46413883236430091570208500028, −3.23088674870732935194609608941, −2.50994638478549492939056655724, −1.65461411133275513613208801046,
1.65461411133275513613208801046, 2.50994638478549492939056655724, 3.23088674870732935194609608941, 4.46413883236430091570208500028, 5.03027375920784564217785266273, 5.86126892323299065677908591135, 6.24723087762616691306285278455, 6.78489189350551972813060491708, 7.41365440531891275049274903808, 8.197442728315647154964018015629, 8.718569936936276174746017611956, 9.339836646165358185420325475545, 9.737481940209185346237104800034, 10.28353392795342278987908972818, 10.82553789347932959955315112147
