| L(s) = 1 | − 4·7-s + 2·9-s − 12·25-s + 64·43-s + 16·49-s − 8·63-s − 48·67-s − 8·79-s + 2·81-s − 64·109-s + 24·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 12·169-s + 173-s + 48·175-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 2/3·9-s − 2.39·25-s + 9.75·43-s + 16/7·49-s − 1.00·63-s − 5.86·67-s − 0.900·79-s + 2/9·81-s − 6.13·109-s + 2.18·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.923·169-s + 0.0760·173-s + 3.62·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8920302797\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8920302797\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 2 T^{2} + 2 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | \( ( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| good | 5 | \( ( 1 + 6 T^{2} + 42 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 6 T^{2} + 330 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 28 T^{2} + 502 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 66 T^{2} + 1794 T^{4} - 66 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 40 T^{2} + 846 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 64 T^{2} + 2094 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 96 T^{2} + 4158 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 124 T^{2} + 6934 T^{4} + 124 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 8 T + p T^{2} )^{8} \) |
| 47 | \( ( 1 + 132 T^{2} + 8502 T^{4} + 132 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 128 T^{2} + 8014 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 78 T^{2} + 7650 T^{4} + 78 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 54 T^{2} + 7338 T^{4} - 54 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 76 T^{2} + 1734 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - p T^{2} )^{8} \) |
| 79 | \( ( 1 + 2 T + 142 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 2 T^{2} - 2558 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 244 T^{2} + 29638 T^{4} + 244 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 140 T^{2} + 10390 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 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
−4.35995884517096078722091756071, −4.30436387511883994361305225292, −4.27797895483756676419784234497, −4.20471420665522042644389263079, −4.13451987580876119229409319602, −4.10921504062110761520382640404, −3.70605818764156995028699799670, −3.60668905252298450787985023897, −3.53544647073113482162883013273, −3.30400448619765335452196114913, −2.99992297184171099414807385976, −2.97207669123051630595841140596, −2.89997621229821709263924039760, −2.55549951449093615048391708258, −2.47754371457772736704462385591, −2.43917774901478156456105955391, −2.39070189515421328305050853046, −2.06106508623185349797189227459, −1.70617383462470894143107460107, −1.54012257634711503570281991811, −1.45358080890331479416041109268, −1.07056919577681234930869176654, −0.869451254442353996738550115487, −0.65824260759859155411271439385, −0.15458845594749216891372834598,
0.15458845594749216891372834598, 0.65824260759859155411271439385, 0.869451254442353996738550115487, 1.07056919577681234930869176654, 1.45358080890331479416041109268, 1.54012257634711503570281991811, 1.70617383462470894143107460107, 2.06106508623185349797189227459, 2.39070189515421328305050853046, 2.43917774901478156456105955391, 2.47754371457772736704462385591, 2.55549951449093615048391708258, 2.89997621229821709263924039760, 2.97207669123051630595841140596, 2.99992297184171099414807385976, 3.30400448619765335452196114913, 3.53544647073113482162883013273, 3.60668905252298450787985023897, 3.70605818764156995028699799670, 4.10921504062110761520382640404, 4.13451987580876119229409319602, 4.20471420665522042644389263079, 4.27797895483756676419784234497, 4.30436387511883994361305225292, 4.35995884517096078722091756071
Plot not available for L-functions of degree greater than 10.
