| L(s) = 1 | + 2·9-s + 4·13-s + 8·19-s − 25-s + 20·43-s − 12·47-s + 10·49-s + 12·53-s − 24·59-s + 4·67-s − 5·81-s − 12·83-s − 12·89-s + 12·101-s + 28·103-s + 8·117-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 14·169-s + ⋯ |
| L(s) = 1 | + 2/3·9-s + 1.10·13-s + 1.83·19-s − 1/5·25-s + 3.04·43-s − 1.75·47-s + 10/7·49-s + 1.64·53-s − 3.12·59-s + 0.488·67-s − 5/9·81-s − 1.31·83-s − 1.27·89-s + 1.19·101-s + 2.75·103-s + 0.739·117-s + 2·121-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 − 1.07·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33408400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33408400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.753927031\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.753927031\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 17 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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
−8.616773272986044176881022167148, −7.73276061797015930620379998943, −7.54550727696115930804371067082, −7.27855198407082602559688554469, −7.09424566657196818893143442697, −6.42964571963659536158206486889, −6.03874362819015385246231840984, −5.85326470100329636488713623089, −5.59115023153883955987047102669, −4.85827161112709190911421409391, −4.79908745620068011328590817665, −4.13598502224233865669736387375, −3.97450644175189896761544431501, −3.32920445921698485353125905042, −3.21686565216667850936633096004, −2.54698066709630898551387348849, −2.15625909029528111671861946318, −1.32115725402431437071387720035, −1.24702266708740534038997214756, −0.55378584897632678778180131710,
0.55378584897632678778180131710, 1.24702266708740534038997214756, 1.32115725402431437071387720035, 2.15625909029528111671861946318, 2.54698066709630898551387348849, 3.21686565216667850936633096004, 3.32920445921698485353125905042, 3.97450644175189896761544431501, 4.13598502224233865669736387375, 4.79908745620068011328590817665, 4.85827161112709190911421409391, 5.59115023153883955987047102669, 5.85326470100329636488713623089, 6.03874362819015385246231840984, 6.42964571963659536158206486889, 7.09424566657196818893143442697, 7.27855198407082602559688554469, 7.54550727696115930804371067082, 7.73276061797015930620379998943, 8.616773272986044176881022167148
