| L(s) = 1 | − 4·3-s + 6·9-s − 16·13-s + 2·25-s + 4·27-s + 16·37-s + 64·39-s − 4·49-s − 8·75-s − 37·81-s + 24·83-s + 24·107-s − 64·111-s − 96·117-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 16·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 108·169-s + 173-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 2·9-s − 4.43·13-s + 2/5·25-s + 0.769·27-s + 2.63·37-s + 10.2·39-s − 4/7·49-s − 0.923·75-s − 4.11·81-s + 2.63·83-s + 2.32·107-s − 6.07·111-s − 8.87·117-s + 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.31·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 8.30·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6366850223\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6366850223\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.43678939689454967201466238410, −6.16103682246704684348476870080, −6.08327730587091537403102742913, −6.02125429198556818268826223766, −5.79956355113466671773247379996, −5.27360600366751785401594080367, −5.26764827002638977775951727462, −5.01952055162942933530679123113, −4.94331406557041100144054401688, −4.80610120875662198515325850982, −4.46182943611757777984384949423, −4.34748853158236285323119408948, −4.29370681968692912922022378637, −3.67667091349029810149532658277, −3.32916990494010074871021674523, −3.25749098564292109044279945196, −2.69411563677460750204983311165, −2.59029108507560910914665935113, −2.51342313720341614693626871516, −2.14959076284134416362613747544, −1.82809752596557022559839577463, −1.38611170643453274046476245152, −0.71304166986781221057140016004, −0.61250278737152370601670085154, −0.32304449282992140779406087289,
0.32304449282992140779406087289, 0.61250278737152370601670085154, 0.71304166986781221057140016004, 1.38611170643453274046476245152, 1.82809752596557022559839577463, 2.14959076284134416362613747544, 2.51342313720341614693626871516, 2.59029108507560910914665935113, 2.69411563677460750204983311165, 3.25749098564292109044279945196, 3.32916990494010074871021674523, 3.67667091349029810149532658277, 4.29370681968692912922022378637, 4.34748853158236285323119408948, 4.46182943611757777984384949423, 4.80610120875662198515325850982, 4.94331406557041100144054401688, 5.01952055162942933530679123113, 5.26764827002638977775951727462, 5.27360600366751785401594080367, 5.79956355113466671773247379996, 6.02125429198556818268826223766, 6.08327730587091537403102742913, 6.16103682246704684348476870080, 6.43678939689454967201466238410
