| L(s) = 1 | + 4-s − 9-s + 4·29-s − 36-s − 4·41-s + 49-s + 2·61-s − 64-s + 81-s − 2·89-s + 2·101-s − 2·109-s + 4·116-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 4·164-s + 167-s − 4·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 4-s − 9-s + 4·29-s − 36-s − 4·41-s + 49-s + 2·61-s − 64-s + 81-s − 2·89-s + 2·101-s − 2·109-s + 4·116-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 4·164-s + 167-s − 4·169-s + 173-s + 179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8401873224\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8401873224\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
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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
−7.68543330582300518882892333904, −7.39854027386070060764693101030, −7.20237275803542727261896525076, −7.10872698690888113528102105175, −6.64445824387371851371414029105, −6.51741880116450562745848877659, −6.49499742689639221533950118616, −6.30174284513544197052612291144, −5.89436365706709778425679089108, −5.52760340457769480408736273100, −5.51095066513700780785236470283, −4.94059903892542441235336612623, −4.86003982870407861603028904737, −4.82484768556985802310350256888, −4.44854473068624014863079163004, −3.79940559461868676251154183679, −3.67789542204281055769983056446, −3.63245159183965746595912875875, −2.99190799930552000473215521646, −2.73366311502421252743007216621, −2.57013339079330267561694860797, −2.48801758194578892004379944530, −1.78730139688979551824940677567, −1.49864871645564382853107396410, −0.984672679369739259079610876653,
0.984672679369739259079610876653, 1.49864871645564382853107396410, 1.78730139688979551824940677567, 2.48801758194578892004379944530, 2.57013339079330267561694860797, 2.73366311502421252743007216621, 2.99190799930552000473215521646, 3.63245159183965746595912875875, 3.67789542204281055769983056446, 3.79940559461868676251154183679, 4.44854473068624014863079163004, 4.82484768556985802310350256888, 4.86003982870407861603028904737, 4.94059903892542441235336612623, 5.51095066513700780785236470283, 5.52760340457769480408736273100, 5.89436365706709778425679089108, 6.30174284513544197052612291144, 6.49499742689639221533950118616, 6.51741880116450562745848877659, 6.64445824387371851371414029105, 7.10872698690888113528102105175, 7.20237275803542727261896525076, 7.39854027386070060764693101030, 7.68543330582300518882892333904
