| L(s) = 1 | + 4·7-s − 5·13-s − 14·19-s + 5·25-s + 7·31-s − 20·37-s + 13·43-s + 7·49-s + 13·61-s − 11·67-s + 34·73-s − 17·79-s − 20·91-s + 19·97-s + 7·103-s + 34·109-s + 11·121-s + 127-s + 131-s − 56·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 1.38·13-s − 3.21·19-s + 25-s + 1.25·31-s − 3.28·37-s + 1.98·43-s + 49-s + 1.66·61-s − 1.34·67-s + 3.97·73-s − 1.91·79-s − 2.09·91-s + 1.92·97-s + 0.689·103-s + 3.25·109-s + 121-s + 0.0887·127-s + 0.0873·131-s − 4.85·133-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 944784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 944784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.786333715\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.786333715\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
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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
−10.22462097453571218112343460634, −10.18697061169295571038824450105, −9.162377778935506121976372507627, −8.756652611867347707245320640521, −8.739823279221469597398495088759, −8.124202131819958128206303762430, −7.83153989491991082327661306619, −7.27313939997081265463475314151, −6.80148584373458156962987207243, −6.53627614476559093084639588800, −5.93437472438920757150385202680, −5.23741506028052292590218272383, −4.97173078420817295032312813270, −4.51043623003399902229115609309, −4.19237772252609825951327090288, −3.50557625438951798957335859084, −2.62032001267920941684449414407, −2.12889956551082042608832160227, −1.81224222374119512359537461234, −0.60557878200296286027696848784,
0.60557878200296286027696848784, 1.81224222374119512359537461234, 2.12889956551082042608832160227, 2.62032001267920941684449414407, 3.50557625438951798957335859084, 4.19237772252609825951327090288, 4.51043623003399902229115609309, 4.97173078420817295032312813270, 5.23741506028052292590218272383, 5.93437472438920757150385202680, 6.53627614476559093084639588800, 6.80148584373458156962987207243, 7.27313939997081265463475314151, 7.83153989491991082327661306619, 8.124202131819958128206303762430, 8.739823279221469597398495088759, 8.756652611867347707245320640521, 9.162377778935506121976372507627, 10.18697061169295571038824450105, 10.22462097453571218112343460634
