| L(s) = 1 | − 64·4-s − 324·9-s − 520·11-s + 2.56e3·16-s + 2.67e3·19-s − 7.07e3·29-s − 2.66e4·31-s + 2.07e4·36-s + 2.11e4·41-s + 3.32e4·44-s + 1.05e5·49-s + 3.11e4·59-s − 7.66e4·61-s − 8.19e4·64-s − 1.64e5·71-s − 1.71e5·76-s + 3.42e5·79-s + 6.56e4·81-s + 3.56e5·89-s + 1.68e5·99-s − 1.38e5·101-s + 2.20e5·109-s + 4.52e5·116-s − 8.85e5·121-s + 1.70e6·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 2·4-s − 4/3·9-s − 1.29·11-s + 5/2·16-s + 1.69·19-s − 1.56·29-s − 4.97·31-s + 8/3·36-s + 1.96·41-s + 2.59·44-s + 6.25·49-s + 1.16·59-s − 2.63·61-s − 5/2·64-s − 3.88·71-s − 3.39·76-s + 6.17·79-s + 10/9·81-s + 4.77·89-s + 1.72·99-s − 1.34·101-s + 1.77·109-s + 3.12·116-s − 5.49·121-s + 9.95·124-s + 5.50e−6·127-s + 5.09e−6·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.6364396954\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6364396954\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 3 | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 105162 T^{2} + 5221187235 T^{4} - 3241831047068 p^{2} T^{6} + 3228965148839743149 T^{8} - 3241831047068 p^{12} T^{10} + 5221187235 p^{20} T^{12} - 105162 p^{30} T^{14} + p^{40} T^{16} \) |
| 11 | \( ( 1 + 260 T + 543914 T^{2} + 116274680 T^{3} + 125402312051 T^{4} + 116274680 p^{5} T^{5} + 543914 p^{10} T^{6} + 260 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 13 | \( 1 - 1655748 T^{2} + 1500981496650 T^{4} - 903958782545847728 T^{6} + \)\(39\!\cdots\!19\)\( T^{8} - 903958782545847728 p^{10} T^{10} + 1500981496650 p^{20} T^{12} - 1655748 p^{30} T^{14} + p^{40} T^{16} \) |
| 17 | \( 1 - 7908042 T^{2} + 30056049074555 T^{4} - 72416157334100640852 T^{6} + \)\(12\!\cdots\!69\)\( T^{8} - 72416157334100640852 p^{10} T^{10} + 30056049074555 p^{20} T^{12} - 7908042 p^{30} T^{14} + p^{40} T^{16} \) |
| 19 | \( ( 1 - 1336 T + 4663482 T^{2} - 4396216608 T^{3} + 15365061855595 T^{4} - 4396216608 p^{5} T^{5} + 4663482 p^{10} T^{6} - 1336 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 23 | \( 1 - 5307844 T^{2} - 4852888972278 T^{4} - \)\(16\!\cdots\!92\)\( T^{6} + \)\(34\!\cdots\!95\)\( T^{8} - \)\(16\!\cdots\!92\)\( p^{10} T^{10} - 4852888972278 p^{20} T^{12} - 5307844 p^{30} T^{14} + p^{40} T^{16} \) |
| 29 | \( ( 1 + 3536 T + 55249322 T^{2} + 122394636128 T^{3} + 1388102738838155 T^{4} + 122394636128 p^{5} T^{5} + 55249322 p^{10} T^{6} + 3536 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 31 | \( ( 1 + 13318 T + 136041503 T^{2} + 979128654516 T^{3} + 6148238478015185 T^{4} + 979128654516 p^{5} T^{5} + 136041503 p^{10} T^{6} + 13318 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 37 | \( 1 - 184342580 T^{2} + 16053897260544986 T^{4} - \)\(67\!\cdots\!60\)\( T^{6} + \)\(26\!\cdots\!51\)\( T^{8} - \)\(67\!\cdots\!60\)\( p^{10} T^{10} + 16053897260544986 p^{20} T^{12} - 184342580 p^{30} T^{14} + p^{40} T^{16} \) |
| 41 | \( ( 1 - 10586 T + 243221255 T^{2} - 2215792464944 T^{3} + 1011200944745149 p T^{4} - 2215792464944 p^{5} T^{5} + 243221255 p^{10} T^{6} - 10586 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 43 | \( 1 - 725973970 T^{2} + 268291275799003531 T^{4} - \)\(64\!\cdots\!40\)\( T^{6} + \)\(11\!\cdots\!61\)\( T^{8} - \)\(64\!\cdots\!40\)\( p^{10} T^{10} + 268291275799003531 p^{20} T^{12} - 725973970 p^{30} T^{14} + p^{40} T^{16} \) |
| 47 | \( 1 - 1105003636 T^{2} + 596525439164729882 T^{4} - \)\(21\!\cdots\!08\)\( T^{6} + \)\(56\!\cdots\!95\)\( T^{8} - \)\(21\!\cdots\!08\)\( p^{10} T^{10} + 596525439164729882 p^{20} T^{12} - 1105003636 p^{30} T^{14} + p^{40} T^{16} \) |
| 53 | \( 1 - 1941713170 T^{2} + 1969340174729873331 T^{4} - \)\(13\!\cdots\!40\)\( T^{6} + \)\(64\!\cdots\!61\)\( T^{8} - \)\(13\!\cdots\!40\)\( p^{10} T^{10} + 1969340174729873331 p^{20} T^{12} - 1941713170 p^{30} T^{14} + p^{40} T^{16} \) |
| 59 | \( ( 1 - 15590 T + 1982857631 T^{2} - 21599962598780 T^{3} + 1829829882901049561 T^{4} - 21599962598780 p^{5} T^{5} + 1982857631 p^{10} T^{6} - 15590 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 61 | \( ( 1 + 38310 T + 3642692119 T^{2} + 94859357653080 T^{3} + 4723185010974952461 T^{4} + 94859357653080 p^{5} T^{5} + 3642692119 p^{10} T^{6} + 38310 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 67 | \( 1 - 7607192312 T^{2} + 28021646918105287740 T^{4} - \)\(65\!\cdots\!12\)\( T^{6} + \)\(10\!\cdots\!34\)\( T^{8} - \)\(65\!\cdots\!12\)\( p^{10} T^{10} + 28021646918105287740 p^{20} T^{12} - 7607192312 p^{30} T^{14} + p^{40} T^{16} \) |
| 71 | \( ( 1 + 82410 T + 8354416219 T^{2} + 425824660979880 T^{3} + 23696889318453224361 T^{4} + 425824660979880 p^{5} T^{5} + 8354416219 p^{10} T^{6} + 82410 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 73 | \( 1 - 9683148578 T^{2} + 40694859232333067315 T^{4} - \)\(10\!\cdots\!88\)\( T^{6} + \)\(21\!\cdots\!69\)\( T^{8} - \)\(10\!\cdots\!88\)\( p^{10} T^{10} + 40694859232333067315 p^{20} T^{12} - 9683148578 p^{30} T^{14} + p^{40} T^{16} \) |
| 79 | \( ( 1 - 171228 T + 19884267880 T^{2} - 1602486319915548 T^{3} + \)\(10\!\cdots\!74\)\( T^{4} - 1602486319915548 p^{5} T^{5} + 19884267880 p^{10} T^{6} - 171228 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 83 | \( 1 - 17063303138 T^{2} + \)\(14\!\cdots\!75\)\( T^{4} - \)\(87\!\cdots\!28\)\( T^{6} + \)\(39\!\cdots\!29\)\( T^{8} - \)\(87\!\cdots\!28\)\( p^{10} T^{10} + \)\(14\!\cdots\!75\)\( p^{20} T^{12} - 17063303138 p^{30} T^{14} + p^{40} T^{16} \) |
| 89 | \( ( 1 - 178428 T + 31254961130 T^{2} - 3082412386202448 T^{3} + \)\(28\!\cdots\!99\)\( T^{4} - 3082412386202448 p^{5} T^{5} + 31254961130 p^{10} T^{6} - 178428 p^{15} T^{7} + p^{20} T^{8} )^{2} \) |
| 97 | \( 1 - 30681460730 T^{2} + \)\(59\!\cdots\!71\)\( T^{4} - \)\(77\!\cdots\!60\)\( T^{6} + \)\(77\!\cdots\!81\)\( T^{8} - \)\(77\!\cdots\!60\)\( p^{10} T^{10} + \)\(59\!\cdots\!71\)\( p^{20} T^{12} - 30681460730 p^{30} T^{14} + p^{40} T^{16} \) |
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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
−3.77076043961235847334109960665, −3.42379090531374770570377563377, −3.39186996894010714210105553941, −3.37755249410248185754984467727, −3.35909463340344449548127645430, −3.06984646417999923143009249201, −2.76288312965693640229852440963, −2.64005710159202693414546729765, −2.54387583646878739617362751290, −2.51025480436143104623386172465, −2.25686744838721764099584245649, −2.22925109238838408996057144580, −1.86169907403401932439686006236, −1.86071931666797913179207471521, −1.74591186949367820894593040423, −1.62939341122179239383406408836, −1.13748159845592964073363326818, −1.11960433222787555896735680284, −0.934396712302826411591210086535, −0.893448573426932244904952051980, −0.67351343742547103024567665079, −0.53777818869197458701773540603, −0.32894914401414981218378876867, −0.23018586174338685816865794811, −0.088674241029845886061201469914,
0.088674241029845886061201469914, 0.23018586174338685816865794811, 0.32894914401414981218378876867, 0.53777818869197458701773540603, 0.67351343742547103024567665079, 0.893448573426932244904952051980, 0.934396712302826411591210086535, 1.11960433222787555896735680284, 1.13748159845592964073363326818, 1.62939341122179239383406408836, 1.74591186949367820894593040423, 1.86071931666797913179207471521, 1.86169907403401932439686006236, 2.22925109238838408996057144580, 2.25686744838721764099584245649, 2.51025480436143104623386172465, 2.54387583646878739617362751290, 2.64005710159202693414546729765, 2.76288312965693640229852440963, 3.06984646417999923143009249201, 3.35909463340344449548127645430, 3.37755249410248185754984467727, 3.39186996894010714210105553941, 3.42379090531374770570377563377, 3.77076043961235847334109960665
Plot not available for L-functions of degree greater than 10.
