| L(s) = 1 | − 9-s − 2·17-s − 2·25-s + 2·29-s − 49-s + 4·53-s − 2·61-s + 2·101-s + 2·113-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2·153-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 9-s − 2·17-s − 2·25-s + 2·29-s − 49-s + 4·53-s − 2·61-s + 2·101-s + 2·113-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2·153-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7311616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7311616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8512001117\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8512001117\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_1$ | \( ( 1 - T )^{4} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 97 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
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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
−8.934525833536397827144386193716, −8.903620517746756231940326389792, −8.566590558113237267420064037320, −8.105260927927771572910405022195, −7.78811723468810327963535304823, −7.29281439323160589133365953955, −6.91884297073411792322350202849, −6.49225603545020559720428677849, −6.11224336856547709857882399584, −5.87583401166041696552095288580, −5.37507562109035485309504188956, −4.86006029836980343839606603151, −4.49069108569335283089668351437, −4.09906706481951932644294327235, −3.64496634038730119716715207080, −3.04845471591141126975645567386, −2.52501102719438977868290239860, −2.23904302000369813836553508180, −1.62264960455214681806899407702, −0.57844933889258668595899675297,
0.57844933889258668595899675297, 1.62264960455214681806899407702, 2.23904302000369813836553508180, 2.52501102719438977868290239860, 3.04845471591141126975645567386, 3.64496634038730119716715207080, 4.09906706481951932644294327235, 4.49069108569335283089668351437, 4.86006029836980343839606603151, 5.37507562109035485309504188956, 5.87583401166041696552095288580, 6.11224336856547709857882399584, 6.49225603545020559720428677849, 6.91884297073411792322350202849, 7.29281439323160589133365953955, 7.78811723468810327963535304823, 8.105260927927771572910405022195, 8.566590558113237267420064037320, 8.903620517746756231940326389792, 8.934525833536397827144386193716
