| L(s) = 1 | − 4·4-s − 4·7-s − 2·9-s + 10·16-s − 12·25-s + 16·28-s + 8·36-s + 56·43-s + 16·49-s + 8·63-s − 20·64-s − 96·67-s − 88·79-s + 2·81-s + 48·100-s + 8·109-s − 40·112-s − 4·121-s + 127-s + 131-s + 137-s + 139-s − 20·144-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 2·4-s − 1.51·7-s − 2/3·9-s + 5/2·16-s − 2.39·25-s + 3.02·28-s + 4/3·36-s + 8.53·43-s + 16/7·49-s + 1.00·63-s − 5/2·64-s − 11.7·67-s − 9.90·79-s + 2/9·81-s + 24/5·100-s + 0.766·109-s − 3.77·112-s − 0.363·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5/3·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5478210966\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5478210966\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{4} \) |
| 3 | \( 1 + 2 T^{2} + 2 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | \( ( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | \( ( 1 + T^{2} )^{4} \) |
| good | 5 | \( ( 1 + 6 T^{2} + 42 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 6 T^{2} + 330 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 40 T^{2} + 910 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 2 T^{2} + 706 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 104 T^{2} + 4558 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 72 T^{2} + 4590 T^{4} + 72 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 14 T + 118 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + 96 T^{2} + 6654 T^{4} + 96 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 30 T^{2} + 7170 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 230 T^{2} + 20650 T^{4} - 230 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 12 T + p T^{2} )^{8} \) |
| 71 | \( ( 1 + 40 T^{2} + 4974 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 108 T^{2} + 13302 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 22 T + 262 T^{2} + 22 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 258 T^{2} + 30402 T^{4} + 258 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 132 T^{2} + 15846 T^{4} + 132 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 348 T^{2} + 48822 T^{4} - 348 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.72346405719103189860120860320, −4.48170458967583538841152280874, −4.37695202404596451734527674660, −4.36334750987217848319265546399, −4.33144346201873242825829053756, −4.23029932431787326371095825722, −4.22597905497841157865565864010, −3.96283407624246029131825622045, −3.94924233610711625766949691895, −3.42600699079087267984966721608, −3.39144784555083862145230364757, −3.24605979323439215602511847597, −2.97686631486887885778740008400, −2.95729732565828514587034641264, −2.73745495553240124395705694684, −2.60828087285390021494490273502, −2.58029276408909777813757558934, −2.44312017933496780989630380696, −1.65627216963011825020088320753, −1.62715145510112162612364151887, −1.51923814776803832283789348881, −1.51369248127140402839567610572, −0.66524824426725291495190833884, −0.60386274907848911777396302321, −0.25954109118765686491711803711,
0.25954109118765686491711803711, 0.60386274907848911777396302321, 0.66524824426725291495190833884, 1.51369248127140402839567610572, 1.51923814776803832283789348881, 1.62715145510112162612364151887, 1.65627216963011825020088320753, 2.44312017933496780989630380696, 2.58029276408909777813757558934, 2.60828087285390021494490273502, 2.73745495553240124395705694684, 2.95729732565828514587034641264, 2.97686631486887885778740008400, 3.24605979323439215602511847597, 3.39144784555083862145230364757, 3.42600699079087267984966721608, 3.94924233610711625766949691895, 3.96283407624246029131825622045, 4.22597905497841157865565864010, 4.23029932431787326371095825722, 4.33144346201873242825829053756, 4.36334750987217848319265546399, 4.37695202404596451734527674660, 4.48170458967583538841152280874, 4.72346405719103189860120860320
Plot not available for L-functions of degree greater than 10.
