| L(s) = 1 | − 176·7-s + 1.37e3·13-s − 5.88e3·19-s + 4.59e3·25-s + 4.14e3·31-s + 3.35e3·37-s + 3.83e4·43-s + 5.91e4·49-s + 1.20e5·61-s + 1.07e5·67-s − 1.84e5·73-s − 5.02e3·79-s − 2.41e5·91-s + 3.31e5·97-s + 1.65e5·103-s − 5.16e5·109-s + 4.38e5·121-s + 127-s + 131-s + 1.03e6·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 1.35·7-s + 2.25·13-s − 3.74·19-s + 1.47·25-s + 0.774·31-s + 0.402·37-s + 3.16·43-s + 3.52·49-s + 4.13·61-s + 2.92·67-s − 4.05·73-s − 0.0905·79-s − 3.05·91-s + 3.58·97-s + 1.53·103-s − 4.16·109-s + 2.72·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 5.07·133-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.810790353\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.810790353\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 4598 T^{2} + 5170153 T^{4} + 16367955802 T^{6} - 46741161157724 T^{8} + 16367955802 p^{10} T^{10} + 5170153 p^{20} T^{12} - 4598 p^{30} T^{14} + p^{40} T^{16} \) |
| 7 | \( ( 1 + 88 T - 17978 T^{2} - 694496 T^{3} + 248992627 T^{4} - 694496 p^{5} T^{5} - 17978 p^{10} T^{6} + 88 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 11 | \( 1 - 438068 T^{2} + 98987467066 T^{4} - 17978861089360208 T^{6} + \)\(29\!\cdots\!39\)\( T^{8} - 17978861089360208 p^{10} T^{10} + 98987467066 p^{20} T^{12} - 438068 p^{30} T^{14} + p^{40} T^{16} \) |
| 13 | \( ( 1 - 686 T + 91933 T^{2} + 249651178 T^{3} - 169201824908 T^{4} + 249651178 p^{5} T^{5} + 91933 p^{10} T^{6} - 686 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 17 | \( ( 1 + 2188094 T^{2} + 14580022371 p^{2} T^{4} + 2188094 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 19 | \( ( 1 + 1472 T + 3832962 T^{2} + 1472 p^{5} T^{3} + p^{10} T^{4} )^{4} \) |
| 23 | \( 1 - 19977668 T^{2} + 223327114491370 T^{4} - \)\(18\!\cdots\!08\)\( T^{6} + \)\(12\!\cdots\!39\)\( T^{8} - \)\(18\!\cdots\!08\)\( p^{10} T^{10} + 223327114491370 p^{20} T^{12} - 19977668 p^{30} T^{14} + p^{40} T^{16} \) |
| 29 | \( 1 - 68023478 T^{2} + 2675877659197513 T^{4} - \)\(75\!\cdots\!82\)\( T^{6} + \)\(16\!\cdots\!56\)\( T^{8} - \)\(75\!\cdots\!82\)\( p^{10} T^{10} + 2675877659197513 p^{20} T^{12} - 68023478 p^{30} T^{14} + p^{40} T^{16} \) |
| 31 | \( ( 1 - 2072 T - 53684606 T^{2} - 1490779136 T^{3} + 2418885633296515 T^{4} - 1490779136 p^{5} T^{5} - 53684606 p^{10} T^{6} - 2072 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 37 | \( ( 1 - 838 T + 86490063 T^{2} - 838 p^{5} T^{3} + p^{10} T^{4} )^{4} \) |
| 41 | \( 1 - 451482308 T^{2} + 126049105123755754 T^{4} - \)\(22\!\cdots\!64\)\( T^{6} + \)\(31\!\cdots\!35\)\( T^{8} - \)\(22\!\cdots\!64\)\( p^{10} T^{10} + 126049105123755754 p^{20} T^{12} - 451482308 p^{30} T^{14} + p^{40} T^{16} \) |
| 43 | \( ( 1 - 19160 T + 77636542 T^{2} + 87136384480 T^{3} + 8213230553542315 T^{4} + 87136384480 p^{5} T^{5} + 77636542 p^{10} T^{6} - 19160 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 47 | \( 1 - 367391324 T^{2} - 412027407735158 T^{4} - \)\(11\!\cdots\!64\)\( T^{6} + \)\(84\!\cdots\!31\)\( T^{8} - \)\(11\!\cdots\!64\)\( p^{10} T^{10} - 412027407735158 p^{20} T^{12} - 367391324 p^{30} T^{14} + p^{40} T^{16} \) |
| 53 | \( ( 1 + 728451284 T^{2} + 478272292253360790 T^{4} + 728451284 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 59 | \( 1 - 1998785708 T^{2} + 33889526131905422 p T^{4} - \)\(19\!\cdots\!12\)\( T^{6} + \)\(16\!\cdots\!91\)\( T^{8} - \)\(19\!\cdots\!12\)\( p^{10} T^{10} + 33889526131905422 p^{21} T^{12} - 1998785708 p^{30} T^{14} + p^{40} T^{16} \) |
| 61 | \( ( 1 - 60098 T + 1517177101 T^{2} - 24363723250298 T^{3} + 623487594358287604 T^{4} - 24363723250298 p^{5} T^{5} + 1517177101 p^{10} T^{6} - 60098 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 67 | \( ( 1 - 53648 T - 516106658 T^{2} - 37229799341504 T^{3} + 5812048946495544667 T^{4} - 37229799341504 p^{5} T^{5} - 516106658 p^{10} T^{6} - 53648 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 71 | \( ( 1 - 65314012 T^{2} + 4689386407931107110 T^{4} - 65314012 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 73 | \( ( 1 + 46190 T + 3864348579 T^{2} + 46190 p^{5} T^{3} + p^{10} T^{4} )^{4} \) |
| 79 | \( ( 1 + 2512 T + 2007083398 T^{2} - 20485073762624 T^{3} - 5484599137678910141 T^{4} - 20485073762624 p^{5} T^{5} + 2007083398 p^{10} T^{6} + 2512 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 83 | \( 1 - 9492423500 T^{2} + 45164047572639574234 T^{4} - \)\(13\!\cdots\!00\)\( T^{6} + \)\(38\!\cdots\!55\)\( T^{8} - \)\(13\!\cdots\!00\)\( p^{10} T^{10} + 45164047572639574234 p^{20} T^{12} - 9492423500 p^{30} T^{14} + p^{40} T^{16} \) |
| 89 | \( ( 1 - 680992978 T^{2} + 62238193382785937523 T^{4} - 680992978 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 97 | \( ( 1 - 165932 T + 3532157482 T^{2} - 1132749836085296 T^{3} + \)\(26\!\cdots\!47\)\( T^{4} - 1132749836085296 p^{5} T^{5} + 3532157482 p^{10} T^{6} - 165932 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 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
−4.21229287150190913601486722893, −4.20961639346544975920583244829, −3.99966904428422343955317596635, −3.79627472603729119169814782979, −3.63328961242243137607543314512, −3.59857894658196527362566455898, −3.45092063653091415107254840107, −3.05620757574006729287926451160, −2.94905959460982743486227069054, −2.93241872342433171625864016905, −2.66319219940494781372709923973, −2.52798366665871302116274935123, −2.27278824058846695188104916839, −2.20757585188168233958765212317, −2.10412150916265641863992703363, −1.90990548677040428349000730897, −1.69020618340389596337615593999, −1.35995970193104602974905674250, −1.24603814142387106688623144229, −0.919224639004133805424731257948, −0.76998266651925312277524973329, −0.67022814886186390655529629049, −0.61786159403093408433136847688, −0.46912113459157236045159016610, −0.10669707428854922091381041864,
0.10669707428854922091381041864, 0.46912113459157236045159016610, 0.61786159403093408433136847688, 0.67022814886186390655529629049, 0.76998266651925312277524973329, 0.919224639004133805424731257948, 1.24603814142387106688623144229, 1.35995970193104602974905674250, 1.69020618340389596337615593999, 1.90990548677040428349000730897, 2.10412150916265641863992703363, 2.20757585188168233958765212317, 2.27278824058846695188104916839, 2.52798366665871302116274935123, 2.66319219940494781372709923973, 2.93241872342433171625864016905, 2.94905959460982743486227069054, 3.05620757574006729287926451160, 3.45092063653091415107254840107, 3.59857894658196527362566455898, 3.63328961242243137607543314512, 3.79627472603729119169814782979, 3.99966904428422343955317596635, 4.20961639346544975920583244829, 4.21229287150190913601486722893
Plot not available for L-functions of degree greater than 10.
