| L(s) = 1 | + 2·11-s + 16·13-s + 36·23-s − 177·25-s + 292·37-s + 212·47-s − 613·49-s + 40·59-s + 816·61-s − 40·71-s − 1.18e3·73-s + 690·83-s − 2.48e3·97-s + 2.60e3·107-s + 2.85e3·109-s − 2.65e3·121-s + 127-s + 131-s + 137-s + 139-s + 32·143-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4.20e3·169-s + ⋯ |
| L(s) = 1 | + 0.0548·11-s + 0.341·13-s + 0.326·23-s − 1.41·25-s + 1.29·37-s + 0.657·47-s − 1.78·49-s + 0.0882·59-s + 1.71·61-s − 0.0668·71-s − 1.89·73-s + 0.912·83-s − 2.59·97-s + 2.35·107-s + 2.50·109-s − 1.99·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.0187·143-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 1.91·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.809820683\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.809820683\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( 1 + 177 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 613 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 8 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 686 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 590 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 18 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 34470 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 50749 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 146 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 130542 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 109666 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 106 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 208329 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 20 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 408 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 243826 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 20 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 591 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 518878 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 345 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 1360590 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 1241 T + p^{3} T^{2} )^{2} \) |
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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
−9.076711312530352854120303407241, −8.843011605981752503006544406842, −8.272090543024788946038829585922, −8.053354183754915996755404035190, −7.51361326868520746330958371068, −7.34543993210415231045304795925, −6.57680883814183299171121143479, −6.48986362485961314301649082726, −5.89866269258336097386690551023, −5.55657509747208768155164280030, −5.14736816795559432923617116088, −4.54664667434729156126003892844, −4.10359827245248164532859234824, −3.85648373260802052324975502822, −2.98719070358183183029616015365, −2.94642202039581870532625287037, −1.96129365341892950362085775299, −1.77910959052758996853578676943, −0.900721613741799595041368723147, −0.42794627929417515242331361900,
0.42794627929417515242331361900, 0.900721613741799595041368723147, 1.77910959052758996853578676943, 1.96129365341892950362085775299, 2.94642202039581870532625287037, 2.98719070358183183029616015365, 3.85648373260802052324975502822, 4.10359827245248164532859234824, 4.54664667434729156126003892844, 5.14736816795559432923617116088, 5.55657509747208768155164280030, 5.89866269258336097386690551023, 6.48986362485961314301649082726, 6.57680883814183299171121143479, 7.34543993210415231045304795925, 7.51361326868520746330958371068, 8.053354183754915996755404035190, 8.272090543024788946038829585922, 8.843011605981752503006544406842, 9.076711312530352854120303407241
