| L(s) = 1 | − 4·2-s + 10·4-s + 4·7-s − 20·8-s − 2·9-s − 16·14-s + 35·16-s + 8·18-s + 40·28-s − 4·29-s − 56·32-s − 20·36-s + 8·37-s + 8·47-s + 8·49-s − 80·56-s + 16·58-s − 16·61-s − 8·63-s + 84·64-s − 16·67-s + 40·72-s − 20·73-s − 32·74-s + 3·81-s + 40·83-s − 32·94-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 5·4-s + 1.51·7-s − 7.07·8-s − 2/3·9-s − 4.27·14-s + 35/4·16-s + 1.88·18-s + 7.55·28-s − 0.742·29-s − 9.89·32-s − 3.33·36-s + 1.31·37-s + 1.16·47-s + 8/7·49-s − 10.6·56-s + 2.10·58-s − 2.04·61-s − 1.00·63-s + 21/2·64-s − 1.95·67-s + 4.71·72-s − 2.34·73-s − 3.71·74-s + 1/3·81-s + 4.39·83-s − 3.30·94-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.06053172922\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06053172922\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 926 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 48 T^{2} + 1246 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 16 T^{2} - 178 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 2 T + 46 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 4 T + 26 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 28 T^{2} + 230 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 9974 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 8 T + 98 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 172 T^{2} + 16646 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 10 T + 158 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 20 T + 214 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 6374 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 10 T + 206 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.87846381565947630257672808766, −6.27289484199541517683319532887, −6.01442907924315208823330848514, −5.94858988028833324703397323805, −5.93358768970707671931572586364, −5.77755548314899774130532291369, −5.42534779433789194346228609953, −4.98771780387529298047014204053, −4.93904892722958628854237924786, −4.53345093712428492546111559051, −4.41610914226553336469663320287, −4.27074843773739220350446643137, −3.73636232301602977060119511777, −3.50643149976493371832692449099, −3.25938788820596896153586906928, −3.03544420817315915139565531761, −2.62520140197357839201612056485, −2.54973356769017112653337513213, −2.17756307531788737357164183128, −1.89558761345637549971403176455, −1.77283403543852566158658510482, −1.20199719115249094075275423368, −1.18914299901648457914791341008, −0.789476706413939049664863182814, −0.079692439946965251632889828648,
0.079692439946965251632889828648, 0.789476706413939049664863182814, 1.18914299901648457914791341008, 1.20199719115249094075275423368, 1.77283403543852566158658510482, 1.89558761345637549971403176455, 2.17756307531788737357164183128, 2.54973356769017112653337513213, 2.62520140197357839201612056485, 3.03544420817315915139565531761, 3.25938788820596896153586906928, 3.50643149976493371832692449099, 3.73636232301602977060119511777, 4.27074843773739220350446643137, 4.41610914226553336469663320287, 4.53345093712428492546111559051, 4.93904892722958628854237924786, 4.98771780387529298047014204053, 5.42534779433789194346228609953, 5.77755548314899774130532291369, 5.93358768970707671931572586364, 5.94858988028833324703397323805, 6.01442907924315208823330848514, 6.27289484199541517683319532887, 6.87846381565947630257672808766
