| L(s) = 1 | + 4·3-s + 4·7-s + 8·9-s − 16-s + 16·21-s + 12·27-s − 8·31-s − 24·37-s − 24·43-s − 4·48-s + 8·49-s − 24·61-s + 32·63-s + 16·67-s + 20·73-s + 23·81-s − 32·93-s − 12·97-s − 4·103-s − 96·111-s − 4·112-s + 40·121-s + 127-s − 96·129-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 1.51·7-s + 8/3·9-s − 1/4·16-s + 3.49·21-s + 2.30·27-s − 1.43·31-s − 3.94·37-s − 3.65·43-s − 0.577·48-s + 8/7·49-s − 3.07·61-s + 4.03·63-s + 1.95·67-s + 2.34·73-s + 23/9·81-s − 3.31·93-s − 1.21·97-s − 0.394·103-s − 9.11·111-s − 0.377·112-s + 3.63·121-s + 0.0887·127-s − 8.45·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.125778776\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.125778776\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )( 1 + 16 T^{2} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 158 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )( 1 + 56 T^{2} + p^{2} T^{4} ) \) |
| 59 | $C_2^2$ | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 13294 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 170 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−9.746015786771890271875642877818, −8.985692380732752015401268507962, −8.957241834901620649346050735535, −8.716489224863330776250912338929, −8.549842992853152009203692468559, −8.142243146865085970514718100456, −8.061669573307020052751037092460, −7.88489972593363744207462555435, −7.43143313213841432231629025837, −6.99471582093287865796436715133, −6.93392825734304078433315994792, −6.73482976201572640241325291537, −6.20138554022249356674161500386, −5.59018768909418191554155435772, −5.34819308263600551574721031375, −5.00830344452743109570541413242, −4.79237144463857601788691152308, −4.38952672316284747988970119318, −3.85518667600551828575971006169, −3.49215922874212530078715879765, −3.19366268773046050282275955484, −3.07132677816928570787902344062, −1.99628045672654958149132205408, −1.95237266924233552958915165429, −1.69579203176320333609197701026,
1.69579203176320333609197701026, 1.95237266924233552958915165429, 1.99628045672654958149132205408, 3.07132677816928570787902344062, 3.19366268773046050282275955484, 3.49215922874212530078715879765, 3.85518667600551828575971006169, 4.38952672316284747988970119318, 4.79237144463857601788691152308, 5.00830344452743109570541413242, 5.34819308263600551574721031375, 5.59018768909418191554155435772, 6.20138554022249356674161500386, 6.73482976201572640241325291537, 6.93392825734304078433315994792, 6.99471582093287865796436715133, 7.43143313213841432231629025837, 7.88489972593363744207462555435, 8.061669573307020052751037092460, 8.142243146865085970514718100456, 8.549842992853152009203692468559, 8.716489224863330776250912338929, 8.957241834901620649346050735535, 8.985692380732752015401268507962, 9.746015786771890271875642877818
