| L(s) = 1 | − 4·4-s − 48·16-s + 54·17-s + 114·23-s − 58·25-s + 138·29-s + 170·43-s − 179·49-s − 852·53-s − 34·61-s + 448·64-s − 216·68-s − 2.48e3·79-s − 456·92-s + 232·100-s + 3.91e3·101-s − 3.71e3·103-s + 510·107-s − 822·113-s − 552·116-s − 2.15e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 3/4·16-s + 0.770·17-s + 1.03·23-s − 0.463·25-s + 0.883·29-s + 0.602·43-s − 0.521·49-s − 2.20·53-s − 0.0713·61-s + 7/8·64-s − 0.385·68-s − 3.54·79-s − 0.516·92-s + 0.231·100-s + 3.85·101-s − 3.55·103-s + 0.460·107-s − 0.684·113-s − 0.441·116-s − 1.61·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.094263876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.094263876\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + p^{2} T^{2} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 58 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 179 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2155 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 27 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 5915 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 57 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 69 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 54290 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 99719 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 16745 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 85 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 90034 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 426 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 410395 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 17 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 574451 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 375115 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 231166 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 1244 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 962026 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 1315951 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 300239 T^{2} + p^{6} T^{4} \) |
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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
−9.218883539742072980589050618652, −8.962254975805156522841103640230, −8.485118458944135114586963788419, −8.240642238787495625090132914099, −7.62468029585644588431638466048, −7.37844690666233671496484335042, −6.94479167643991053400292848981, −6.27710749166735797984388683560, −6.23500934074067587912894670163, −5.51642712471554735263890513365, −5.10027356250848282585715401493, −4.75200509128778970915759694154, −4.26461854956984945155441710309, −3.89151148736981725006837414788, −3.04593588660560850927418347817, −3.01236119733689018606869116963, −2.22929072781959910677456522098, −1.51608997029439906082986117727, −1.05552906967875329890587494365, −0.25975404239308756247164309666,
0.25975404239308756247164309666, 1.05552906967875329890587494365, 1.51608997029439906082986117727, 2.22929072781959910677456522098, 3.01236119733689018606869116963, 3.04593588660560850927418347817, 3.89151148736981725006837414788, 4.26461854956984945155441710309, 4.75200509128778970915759694154, 5.10027356250848282585715401493, 5.51642712471554735263890513365, 6.23500934074067587912894670163, 6.27710749166735797984388683560, 6.94479167643991053400292848981, 7.37844690666233671496484335042, 7.62468029585644588431638466048, 8.240642238787495625090132914099, 8.485118458944135114586963788419, 8.962254975805156522841103640230, 9.218883539742072980589050618652
