| L(s) = 1 | − 8·2-s + 36·4-s − 120·8-s − 5·9-s + 330·16-s + 40·18-s + 6·25-s − 792·32-s − 180·36-s − 44·43-s − 48·50-s + 1.71e3·64-s + 16·71-s + 600·72-s − 52·79-s + 9·81-s + 352·86-s + 216·100-s + 44·121-s + 127-s − 3.43e3·128-s + 131-s + 137-s + 139-s − 128·142-s − 1.65e3·144-s + 149-s + ⋯ |
| L(s) = 1 | − 5.65·2-s + 18·4-s − 42.4·8-s − 5/3·9-s + 82.5·16-s + 9.42·18-s + 6/5·25-s − 140.·32-s − 30·36-s − 6.70·43-s − 6.78·50-s + 214.5·64-s + 1.89·71-s + 70.7·72-s − 5.85·79-s + 81-s + 37.9·86-s + 21.5·100-s + 4·121-s + 0.0887·127-s − 303.·128-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 10.7·142-s − 137.5·144-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.06259537083\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06259537083\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T )^{8} \) |
| 3 | \( 1 + 5 T^{2} + 16 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | \( 1 - 34 T^{4} + p^{4} T^{8} \) |
| 13 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 5 | \( ( 1 - 3 T^{2} + 44 T^{4} - 3 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 + 61 T^{2} + 1500 T^{4} + 61 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 13 T^{2} + 96 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 2 p T^{2} + 1059 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 53 T^{2} + 1716 T^{4} - 53 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 73 T^{2} + 3180 T^{4} + 73 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 54 T^{2} + 2939 T^{4} - 54 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 87 T^{2} + 4256 T^{4} - 87 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 11 T + 108 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 - 59 T^{2} + 3432 T^{4} - 59 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 100 T^{2} + 6006 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 152 T^{2} + 11550 T^{4} - 152 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 170 T^{2} + 14139 T^{4} - 170 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 225 T^{2} + 21428 T^{4} - 225 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 2 T + p T^{2} )^{8} \) |
| 73 | \( ( 1 + 174 T^{2} + 16115 T^{4} + 174 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 13 T + 192 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 264 T^{2} + 31070 T^{4} - 264 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 16 T + p T^{2} )^{4}( 1 + 16 T + p T^{2} )^{4} \) |
| 97 | \( ( 1 + 249 T^{2} + 34112 T^{4} + 249 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.86434962628243197554175576965, −4.64117772006033055295650584685, −4.50382230276382722376555395331, −4.15574509224355772173576616921, −4.11057208566212940316041648361, −4.07801534114046681141860787923, −3.51214933867541675952484142978, −3.40903143010740773321449203538, −3.27329465912881978846715112476, −3.15774551103178879568000635308, −3.15386633984070953913949512485, −3.08087825152918186629949200606, −2.90699640752344210620984506210, −2.53399417894280992286062283581, −2.41651588967210866406652695902, −2.30322536517924624411873183424, −1.94522496819574800944580624884, −1.92501955632641199576481600557, −1.66495823251427008282235398559, −1.59128937374608044902468963796, −1.37665565081363843645827531542, −1.11004299824629714803495968392, −0.73625931991600220644306005172, −0.38195970427260985084010570015, −0.26500124053925320592168741409,
0.26500124053925320592168741409, 0.38195970427260985084010570015, 0.73625931991600220644306005172, 1.11004299824629714803495968392, 1.37665565081363843645827531542, 1.59128937374608044902468963796, 1.66495823251427008282235398559, 1.92501955632641199576481600557, 1.94522496819574800944580624884, 2.30322536517924624411873183424, 2.41651588967210866406652695902, 2.53399417894280992286062283581, 2.90699640752344210620984506210, 3.08087825152918186629949200606, 3.15386633984070953913949512485, 3.15774551103178879568000635308, 3.27329465912881978846715112476, 3.40903143010740773321449203538, 3.51214933867541675952484142978, 4.07801534114046681141860787923, 4.11057208566212940316041648361, 4.15574509224355772173576616921, 4.50382230276382722376555395331, 4.64117772006033055295650584685, 4.86434962628243197554175576965
Plot not available for L-functions of degree greater than 10.
