| L(s) = 1 | + 8·7-s − 3·9-s + 20·17-s + 4·23-s − 6·25-s − 8·31-s + 10·41-s − 12·47-s + 30·49-s − 24·63-s + 24·71-s + 36·73-s − 28·79-s − 56·89-s − 2·97-s − 12·103-s − 12·113-s + 160·119-s − 13·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 60·153-s + 157-s + ⋯ |
| L(s) = 1 | + 3.02·7-s − 9-s + 4.85·17-s + 0.834·23-s − 6/5·25-s − 1.43·31-s + 1.56·41-s − 1.75·47-s + 30/7·49-s − 3.02·63-s + 2.84·71-s + 4.21·73-s − 3.15·79-s − 5.93·89-s − 0.203·97-s − 1.18·103-s − 1.12·113-s + 14.6·119-s − 1.18·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 4.85·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.266786116\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.266786116\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| good | 5 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 13 T^{2} + 48 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )( 1 + 23 T^{2} + p^{2} T^{4} ) \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 2 T - 19 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 5 T - 16 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 - 35 T^{2} - 624 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2^3$ | \( 1 + 117 T^{2} + 10208 T^{4} + 117 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} )( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} ) \) |
| 67 | $C_2^3$ | \( 1 + 125 T^{2} + 11136 T^{4} + 125 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 150 T^{2} + 15611 T^{4} + 150 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−8.364420091481086857033298682894, −8.211342447869230299336353052980, −7.980450472901583478422382616210, −7.941798803842492948921589613286, −7.76504140347977063934648337782, −7.60548936776556772389054915728, −7.07298204724172767356251225288, −6.84576408955375056866623468585, −6.71076027218495004892389958893, −5.83158550300280307657623435153, −5.66957857793815988515310100919, −5.63498325059582975640730499846, −5.57972333927483068417834452888, −5.10559013535950030807782824660, −4.90963042318435002805189605630, −4.68312607265359988240416812973, −4.04041462531186537064021921306, −3.72200493391361527989947976254, −3.70463821838737738958723908868, −3.01510883609160373228349786580, −2.85041523031160849579831399260, −2.29619938113364418040097790512, −1.65162219993007474554059115777, −1.38447067023982624408387265594, −1.05769419556521366685285669188,
1.05769419556521366685285669188, 1.38447067023982624408387265594, 1.65162219993007474554059115777, 2.29619938113364418040097790512, 2.85041523031160849579831399260, 3.01510883609160373228349786580, 3.70463821838737738958723908868, 3.72200493391361527989947976254, 4.04041462531186537064021921306, 4.68312607265359988240416812973, 4.90963042318435002805189605630, 5.10559013535950030807782824660, 5.57972333927483068417834452888, 5.63498325059582975640730499846, 5.66957857793815988515310100919, 5.83158550300280307657623435153, 6.71076027218495004892389958893, 6.84576408955375056866623468585, 7.07298204724172767356251225288, 7.60548936776556772389054915728, 7.76504140347977063934648337782, 7.941798803842492948921589613286, 7.980450472901583478422382616210, 8.211342447869230299336353052980, 8.364420091481086857033298682894
