L(s) = 1 | + 6·9-s + 8·11-s − 8·19-s − 4·29-s + 16·31-s − 12·41-s − 2·49-s + 8·59-s + 4·61-s + 27·81-s + 12·89-s + 48·99-s − 12·101-s + 28·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s − 48·171-s + 173-s + ⋯ |
L(s) = 1 | + 2·9-s + 2.41·11-s − 1.83·19-s − 0.742·29-s + 2.87·31-s − 1.87·41-s − 2/7·49-s + 1.04·59-s + 0.512·61-s + 3·81-s + 1.27·89-s + 4.82·99-s − 1.19·101-s + 2.68·109-s + 2.36·121-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 + 1.69·169-s − 3.67·171-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.525568626\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.525568626\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−9.524562819225867659361458405960, −9.363805917093392064592456493400, −8.754636530346371975029320183850, −8.474820321447292991901261159846, −8.140500694834928447032980225688, −7.53915116080628948119860829468, −7.04271225572366521702658589920, −6.69296861528944708464781370553, −6.50688668432075950827878856500, −6.26692918769151761124674564637, −5.57325222919881844515671315446, −4.75868723536041397725577008107, −4.51510164471184856655057398001, −4.27042601855627920301222850342, −3.63474014915657021108151792694, −3.50313101214119214995756591303, −2.41853587344961721591346306465, −1.88216750789001048552953637940, −1.40211422804760138264466826050, −0.814050284895622139745909516269,
0.814050284895622139745909516269, 1.40211422804760138264466826050, 1.88216750789001048552953637940, 2.41853587344961721591346306465, 3.50313101214119214995756591303, 3.63474014915657021108151792694, 4.27042601855627920301222850342, 4.51510164471184856655057398001, 4.75868723536041397725577008107, 5.57325222919881844515671315446, 6.26692918769151761124674564637, 6.50688668432075950827878856500, 6.69296861528944708464781370553, 7.04271225572366521702658589920, 7.53915116080628948119860829468, 8.140500694834928447032980225688, 8.474820321447292991901261159846, 8.754636530346371975029320183850, 9.363805917093392064592456493400, 9.524562819225867659361458405960