| L(s) = 1 | + 742·9-s − 968·11-s − 2.87e3·19-s − 1.93e4·29-s + 7.20e3·31-s + 3.35e4·41-s + 6.86e4·49-s − 4.90e4·59-s + 432·61-s + 2.39e5·71-s − 2.33e5·79-s + 2.92e5·81-s + 2.77e5·89-s − 7.18e5·99-s + 3.43e5·101-s + 1.82e5·109-s + 5.27e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.66e6·169-s + ⋯ |
| L(s) = 1 | + 3.05·9-s − 2.41·11-s − 1.82·19-s − 4.27·29-s + 1.34·31-s + 3.11·41-s + 4.08·49-s − 1.83·59-s + 0.0148·61-s + 5.63·71-s − 4.20·79-s + 4.94·81-s + 3.70·89-s − 7.36·99-s + 3.35·101-s + 1.47·109-s + 3.27·121-s + 5.50e−6·127-s + 5.09e−6·131-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 + 4.49·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(53.32879999\) |
| \(L(\frac12)\) |
\(\approx\) |
\(53.32879999\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( ( 1 + p^{2} T )^{8} \) |
| good | 3 | \( 1 - 742 T^{2} + 258457 T^{4} - 9330166 p^{2} T^{6} + 303831364 p^{4} T^{8} - 9330166 p^{12} T^{10} + 258457 p^{20} T^{12} - 742 p^{30} T^{14} + p^{40} T^{16} \) |
| 7 | \( 1 - 68630 T^{2} + 2340045121 T^{4} - 52547847650110 T^{6} + 948549217858658756 T^{8} - 52547847650110 p^{10} T^{10} + 2340045121 p^{20} T^{12} - 68630 p^{30} T^{14} + p^{40} T^{16} \) |
| 13 | \( 1 - 1669520 T^{2} + 1366665814396 T^{4} - 748468390006451440 T^{6} + \)\(31\!\cdots\!06\)\( T^{8} - 748468390006451440 p^{10} T^{10} + 1366665814396 p^{20} T^{12} - 1669520 p^{30} T^{14} + p^{40} T^{16} \) |
| 17 | \( 1 - 5723606 T^{2} + 15495476726225 T^{4} - 26626847629338107742 T^{6} + \)\(38\!\cdots\!48\)\( T^{8} - 26626847629338107742 p^{10} T^{10} + 15495476726225 p^{20} T^{12} - 5723606 p^{30} T^{14} + p^{40} T^{16} \) |
| 19 | \( ( 1 + 1438 T + 3972253 T^{2} + 6673206110 T^{3} + 14611825029596 T^{4} + 6673206110 p^{5} T^{5} + 3972253 p^{10} T^{6} + 1438 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 23 | \( 1 - 21076224 T^{2} + 135949633844860 T^{4} + \)\(17\!\cdots\!32\)\( T^{6} - \)\(57\!\cdots\!22\)\( T^{8} + \)\(17\!\cdots\!32\)\( p^{10} T^{10} + 135949633844860 p^{20} T^{12} - 21076224 p^{30} T^{14} + p^{40} T^{16} \) |
| 29 | \( ( 1 + 9688 T + 94492307 T^{2} + 564738036036 T^{3} + 3011755326230184 T^{4} + 564738036036 p^{5} T^{5} + 94492307 p^{10} T^{6} + 9688 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 31 | \( ( 1 - 3604 T + 110319863 T^{2} - 285315545172 T^{3} + 4658421727940624 T^{4} - 285315545172 p^{5} T^{5} + 110319863 p^{10} T^{6} - 3604 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 37 | \( 1 - 274025606 T^{2} + 40498651911084025 T^{4} - \)\(41\!\cdots\!42\)\( T^{6} + \)\(32\!\cdots\!48\)\( T^{8} - \)\(41\!\cdots\!42\)\( p^{10} T^{10} + 40498651911084025 p^{20} T^{12} - 274025606 p^{30} T^{14} + p^{40} T^{16} \) |
| 41 | \( ( 1 - 16756 T + 284281748 T^{2} - 3984082765548 T^{3} + 48148074423176694 T^{4} - 3984082765548 p^{5} T^{5} + 284281748 p^{10} T^{6} - 16756 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 43 | \( 1 - 354660944 T^{2} + 76018772552106172 T^{4} - \)\(85\!\cdots\!92\)\( T^{6} + \)\(11\!\cdots\!70\)\( T^{8} - \)\(85\!\cdots\!92\)\( p^{10} T^{10} + 76018772552106172 p^{20} T^{12} - 354660944 p^{30} T^{14} + p^{40} T^{16} \) |
| 47 | \( 1 - 248107808 T^{2} + 57249098692993532 T^{4} - \)\(22\!\cdots\!56\)\( T^{6} + \)\(55\!\cdots\!34\)\( T^{8} - \)\(22\!\cdots\!56\)\( p^{10} T^{10} + 57249098692993532 p^{20} T^{12} - 248107808 p^{30} T^{14} + p^{40} T^{16} \) |
| 53 | \( 1 - 500386134 T^{2} + 236906190719639305 T^{4} - \)\(60\!\cdots\!38\)\( T^{6} + \)\(60\!\cdots\!88\)\( T^{8} - \)\(60\!\cdots\!38\)\( p^{10} T^{10} + 236906190719639305 p^{20} T^{12} - 500386134 p^{30} T^{14} + p^{40} T^{16} \) |
| 59 | \( ( 1 + 24516 T + 2158228000 T^{2} + 31165412137572 T^{3} + 1943461102687507598 T^{4} + 31165412137572 p^{5} T^{5} + 2158228000 p^{10} T^{6} + 24516 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 61 | \( ( 1 - 216 T + 974277827 T^{2} - 28052505674820 T^{3} + 248993122317330216 T^{4} - 28052505674820 p^{5} T^{5} + 974277827 p^{10} T^{6} - 216 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 67 | \( 1 - 7786322312 T^{2} + 29372677916379238300 T^{4} - \)\(69\!\cdots\!72\)\( T^{6} + \)\(11\!\cdots\!54\)\( T^{8} - \)\(69\!\cdots\!72\)\( p^{10} T^{10} + 29372677916379238300 p^{20} T^{12} - 7786322312 p^{30} T^{14} + p^{40} T^{16} \) |
| 71 | \( ( 1 - 119692 T + 7592322975 T^{2} - 379898288546556 T^{3} + 17534301925534876048 T^{4} - 379898288546556 p^{5} T^{5} + 7592322975 p^{10} T^{6} - 119692 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 73 | \( 1 - 14435076592 T^{2} + 94775243290942552732 T^{4} - \)\(36\!\cdots\!44\)\( T^{6} + \)\(93\!\cdots\!34\)\( T^{8} - \)\(36\!\cdots\!44\)\( p^{10} T^{10} + 94775243290942552732 p^{20} T^{12} - 14435076592 p^{30} T^{14} + p^{40} T^{16} \) |
| 79 | \( ( 1 + 116592 T + 14907300728 T^{2} + 1019441217686160 T^{3} + 71794519121254888206 T^{4} + 1019441217686160 p^{5} T^{5} + 14907300728 p^{10} T^{6} + 116592 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 83 | \( 1 - 10729235880 T^{2} + 59504603077799014396 T^{4} - \)\(33\!\cdots\!60\)\( T^{6} + \)\(16\!\cdots\!06\)\( T^{8} - \)\(33\!\cdots\!60\)\( p^{10} T^{10} + 59504603077799014396 p^{20} T^{12} - 10729235880 p^{30} T^{14} + p^{40} T^{16} \) |
| 89 | \( ( 1 - 138554 T + 13106030249 T^{2} - 595454382333786 T^{3} + 38154766942892651844 T^{4} - 595454382333786 p^{5} T^{5} + 13106030249 p^{10} T^{6} - 138554 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 97 | \( 1 - 51799442736 T^{2} + \)\(12\!\cdots\!60\)\( T^{4} - \)\(19\!\cdots\!12\)\( T^{6} + \)\(19\!\cdots\!78\)\( T^{8} - \)\(19\!\cdots\!12\)\( p^{10} T^{10} + \)\(12\!\cdots\!60\)\( p^{20} T^{12} - 51799442736 p^{30} T^{14} + p^{40} T^{16} \) |
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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
−3.39805865769294698898346237151, −3.36585275920933411276381481097, −3.21201642129990051479787944365, −3.02002678789368610017773737177, −2.88498351184459031630546639993, −2.62992476842705655987692953454, −2.62704893140911318476662629433, −2.53219109158624700836245432484, −2.48046503345702591826801698214, −2.10950778439545270249621508385, −2.01598383159565705113344868982, −1.99079004824958380424159754006, −1.80454073242521573492490812644, −1.76064728533867449803158530989, −1.75206879126984824715737000585, −1.49649674483699085474056157612, −1.44551216192266503308748123128, −0.906007650281295389993589023202, −0.870942792100377884350064348820, −0.67134381380497242324439564562, −0.65855713628346524182652877393, −0.60370850862500889294739025568, −0.56684611626940679642350208756, −0.31427650300936950866211058225, −0.24334256357037001636264402209,
0.24334256357037001636264402209, 0.31427650300936950866211058225, 0.56684611626940679642350208756, 0.60370850862500889294739025568, 0.65855713628346524182652877393, 0.67134381380497242324439564562, 0.870942792100377884350064348820, 0.906007650281295389993589023202, 1.44551216192266503308748123128, 1.49649674483699085474056157612, 1.75206879126984824715737000585, 1.76064728533867449803158530989, 1.80454073242521573492490812644, 1.99079004824958380424159754006, 2.01598383159565705113344868982, 2.10950778439545270249621508385, 2.48046503345702591826801698214, 2.53219109158624700836245432484, 2.62704893140911318476662629433, 2.62992476842705655987692953454, 2.88498351184459031630546639993, 3.02002678789368610017773737177, 3.21201642129990051479787944365, 3.36585275920933411276381481097, 3.39805865769294698898346237151
Plot not available for L-functions of degree greater than 10.
