| L(s) = 1 | + 4·3-s + 4·4-s + 10·7-s + 3·9-s + 16·12-s + 10·13-s + 14·16-s + 10·19-s + 40·21-s − 14·25-s − 10·27-s + 40·28-s + 10·31-s + 12·36-s − 6·37-s + 40·39-s + 56·48-s + 31·49-s + 40·52-s + 40·57-s + 20·61-s + 30·63-s + 35·64-s − 4·67-s + 80·73-s − 56·75-s + 40·76-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 2·4-s + 3.77·7-s + 9-s + 4.61·12-s + 2.77·13-s + 7/2·16-s + 2.29·19-s + 8.72·21-s − 2.79·25-s − 1.92·27-s + 7.55·28-s + 1.79·31-s + 2·36-s − 0.986·37-s + 6.40·39-s + 8.08·48-s + 31/7·49-s + 5.54·52-s + 5.29·57-s + 2.56·61-s + 3.77·63-s + 35/8·64-s − 0.488·67-s + 9.36·73-s − 6.46·75-s + 4.58·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(40.87042218\) |
| \(L(\frac12)\) |
\(\approx\) |
\(40.87042218\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 4 T + 13 T^{2} - 10 p T^{3} + 61 T^{4} - 10 p^{2} T^{5} + 13 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - p^{2} T^{2} + p T^{4} + 13 T^{6} - 35 T^{8} + 13 p^{2} T^{10} + p^{5} T^{12} - p^{8} T^{14} + p^{8} T^{16} \) |
| 5 | \( 1 + 14 T^{2} + 51 T^{4} - 356 T^{6} - 3739 T^{8} - 356 p^{2} T^{10} + 51 p^{4} T^{12} + 14 p^{6} T^{14} + p^{8} T^{16} \) |
| 7 | \( ( 1 - 5 T + 22 T^{2} - 85 T^{3} + 229 T^{4} - 85 p T^{5} + 22 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 5 T + 28 T^{2} - 145 T^{3} + 499 T^{4} - 145 p T^{5} + 28 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 - 9 T^{2} + 267 T^{4} - 131 p T^{6} + 32880 T^{8} - 131 p^{3} T^{10} + 267 p^{4} T^{12} - 9 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( ( 1 - 5 T + p T^{2} + 5 p T^{3} - 464 T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 50 T^{2} + 1363 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( 1 - 38 T^{2} + 1523 T^{4} - 1784 p T^{6} + 935125 T^{8} - 1784 p^{3} T^{10} + 1523 p^{4} T^{12} - 38 p^{6} T^{14} + p^{8} T^{16} \) |
| 31 | \( ( 1 - 5 T + 9 T^{2} - 205 T^{3} + 1916 T^{4} - 205 p T^{5} + 9 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 3 T - 28 T^{2} - 195 T^{3} + 451 T^{4} - 195 p T^{5} - 28 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( 1 - 67 T^{2} + 4308 T^{4} - 255209 T^{6} + 10464455 T^{8} - 255209 p^{2} T^{10} + 4308 p^{4} T^{12} - 67 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 - 122 T^{2} + 6919 T^{4} - 122 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 + 50 T^{2} - 969 T^{4} - 128660 T^{6} - 3507499 T^{8} - 128660 p^{2} T^{10} - 969 p^{4} T^{12} + 50 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( 1 + 115 T^{2} + 6051 T^{4} + 72185 T^{6} - 4134064 T^{8} + 72185 p^{2} T^{10} + 6051 p^{4} T^{12} + 115 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( 1 + 107 T^{2} + 9663 T^{4} + 800929 T^{6} + 52808600 T^{8} + 800929 p^{2} T^{10} + 9663 p^{4} T^{12} + 107 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 - 10 T + 111 T^{2} - 1040 T^{3} + 8421 T^{4} - 1040 p T^{5} + 111 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + T + 73 T^{2} + p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( 1 + 162 T^{2} + 5103 T^{4} - 803356 T^{6} - 100375395 T^{8} - 803356 p^{2} T^{10} + 5103 p^{4} T^{12} + 162 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( ( 1 - 40 T + 713 T^{2} - 8040 T^{3} + 72529 T^{4} - 8040 p T^{5} + 713 p^{2} T^{6} - 40 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 5 T + 114 T^{2} + 1705 T^{3} + 6861 T^{4} + 1705 p T^{5} + 114 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 231 T^{2} + 16452 T^{4} - 260413 T^{6} - 14194425 T^{8} - 260413 p^{2} T^{10} + 16452 p^{4} T^{12} - 231 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 266 T^{2} + 31531 T^{4} - 266 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 8 T - 63 T^{2} + 980 T^{3} - 1099 T^{4} + 980 p T^{5} - 63 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 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
−5.05733321284161685411580381677, −5.00890101724616393729936035933, −4.93516015418070658207097511970, −4.67338895095782615932716591257, −4.56368896957343405550427299252, −4.29394588627270085938683186960, −4.04799113302939347672273374240, −3.79651390727933330141179838338, −3.61000651620701535308186458775, −3.55266149209422139289083828757, −3.53896334205812289694650268242, −3.51297166772126340408079594780, −3.31265249617817897010146336773, −3.29357347329436903677743565259, −2.53159056849721288215983833662, −2.46068963127944908602100134710, −2.45117158080334691990042194296, −2.42840956766604302439881000090, −2.23091289566519498040993816797, −2.10850429176321139082450877097, −1.56251381688331186213008137516, −1.49931848346879761141031757041, −1.19316677509028854923619685541, −1.15486120647301263769006598568, −1.11945811773839241963771656961,
1.11945811773839241963771656961, 1.15486120647301263769006598568, 1.19316677509028854923619685541, 1.49931848346879761141031757041, 1.56251381688331186213008137516, 2.10850429176321139082450877097, 2.23091289566519498040993816797, 2.42840956766604302439881000090, 2.45117158080334691990042194296, 2.46068963127944908602100134710, 2.53159056849721288215983833662, 3.29357347329436903677743565259, 3.31265249617817897010146336773, 3.51297166772126340408079594780, 3.53896334205812289694650268242, 3.55266149209422139289083828757, 3.61000651620701535308186458775, 3.79651390727933330141179838338, 4.04799113302939347672273374240, 4.29394588627270085938683186960, 4.56368896957343405550427299252, 4.67338895095782615932716591257, 4.93516015418070658207097511970, 5.00890101724616393729936035933, 5.05733321284161685411580381677
Plot not available for L-functions of degree greater than 10.
