| L(s) = 1 | − 108·9-s − 36·13-s − 372·17-s − 588·29-s − 972·37-s + 6.31e3·41-s + 6.54e3·49-s + 4.98e3·53-s + 1.67e4·61-s + 2.12e4·73-s + 7.29e3·81-s + 9.16e3·89-s + 6.96e3·97-s + 3.17e4·101-s + 3.19e4·109-s + 8.53e4·113-s + 3.88e3·117-s + 5.49e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4.01e4·153-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 4/3·9-s − 0.213·13-s − 1.28·17-s − 0.699·29-s − 0.710·37-s + 3.75·41-s + 2.72·49-s + 1.77·53-s + 4.51·61-s + 3.98·73-s + 10/9·81-s + 1.15·89-s + 0.739·97-s + 3.11·101-s + 2.69·109-s + 6.68·113-s + 0.284·117-s + 3.75·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 1.71·153-s + 4.05e−5·157-s + 3.76e−5·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(160.3861058\) |
| \(L(\frac12)\) |
\(\approx\) |
\(160.3861058\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 6548 T^{2} + 11967700 T^{4} + 6677453716 T^{6} - 1125867613898 p^{2} T^{8} + 6677453716 p^{8} T^{10} + 11967700 p^{16} T^{12} - 6548 p^{24} T^{14} + p^{32} T^{16} \) |
| 11 | \( 1 - 54980 T^{2} + 1546328884 T^{4} - 33475324830140 T^{6} + 570578538692907286 T^{8} - 33475324830140 p^{8} T^{10} + 1546328884 p^{16} T^{12} - 54980 p^{24} T^{14} + p^{32} T^{16} \) |
| 13 | \( ( 1 + 18 T + 74368 T^{2} + 1770606 T^{3} + 2910739710 T^{4} + 1770606 p^{4} T^{5} + 74368 p^{8} T^{6} + 18 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 17 | \( ( 1 + 186 T + 302992 T^{2} + 40066470 T^{3} + 36527815326 T^{4} + 40066470 p^{4} T^{5} + 302992 p^{8} T^{6} + 186 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 19 | \( 1 - 474728 T^{2} + 104295264220 T^{4} - 16618853506493144 T^{6} + \)\(23\!\cdots\!18\)\( T^{8} - 16618853506493144 p^{8} T^{10} + 104295264220 p^{16} T^{12} - 474728 p^{24} T^{14} + p^{32} T^{16} \) |
| 23 | \( 1 - 586664 T^{2} + 197735225308 T^{4} - 44910311339735192 T^{6} + \)\(96\!\cdots\!14\)\( T^{8} - 44910311339735192 p^{8} T^{10} + 197735225308 p^{16} T^{12} - 586664 p^{24} T^{14} + p^{32} T^{16} \) |
| 29 | \( ( 1 + 294 T + 1037228 T^{2} + 108874002 T^{3} + 1033780280214 T^{4} + 108874002 p^{4} T^{5} + 1037228 p^{8} T^{6} + 294 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 31 | \( 1 - 4145240 T^{2} + 9363008695804 T^{4} - 13915854110609702120 T^{6} + \)\(15\!\cdots\!66\)\( T^{8} - 13915854110609702120 p^{8} T^{10} + 9363008695804 p^{16} T^{12} - 4145240 p^{24} T^{14} + p^{32} T^{16} \) |
| 37 | \( ( 1 + 486 T + 3676768 T^{2} + 2804108058 T^{3} + 6896791524414 T^{4} + 2804108058 p^{4} T^{5} + 3676768 p^{8} T^{6} + 486 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 41 | \( ( 1 - 3156 T + 9010532 T^{2} - 18344022060 T^{3} + 31593583024566 T^{4} - 18344022060 p^{4} T^{5} + 9010532 p^{8} T^{6} - 3156 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 43 | \( 1 - 19404488 T^{2} + 176853586778140 T^{4} - \)\(10\!\cdots\!04\)\( T^{6} + \)\(40\!\cdots\!38\)\( T^{8} - \)\(10\!\cdots\!04\)\( p^{8} T^{10} + 176853586778140 p^{16} T^{12} - 19404488 p^{24} T^{14} + p^{32} T^{16} \) |
| 47 | \( 1 - 30505208 T^{2} + 441620068209340 T^{4} - \)\(39\!\cdots\!44\)\( T^{6} + \)\(23\!\cdots\!38\)\( T^{8} - \)\(39\!\cdots\!44\)\( p^{8} T^{10} + 441620068209340 p^{16} T^{12} - 30505208 p^{24} T^{14} + p^{32} T^{16} \) |
| 53 | \( ( 1 - 2490 T + 13935712 T^{2} + 5775982650 T^{3} + 44647373890878 T^{4} + 5775982650 p^{4} T^{5} + 13935712 p^{8} T^{6} - 2490 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 59 | \( 1 - 79327460 T^{2} + 2884299097805044 T^{4} - \)\(63\!\cdots\!80\)\( T^{6} + \)\(92\!\cdots\!66\)\( T^{8} - \)\(63\!\cdots\!80\)\( p^{8} T^{10} + 2884299097805044 p^{16} T^{12} - 79327460 p^{24} T^{14} + p^{32} T^{16} \) |
| 61 | \( ( 1 - 8392 T + 61756540 T^{2} - 257969180536 T^{3} + 1142398355230918 T^{4} - 257969180536 p^{4} T^{5} + 61756540 p^{8} T^{6} - 8392 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 67 | \( 1 - 133180568 T^{2} + 8158394864313340 T^{4} - \)\(30\!\cdots\!04\)\( T^{6} + \)\(73\!\cdots\!58\)\( T^{8} - \)\(30\!\cdots\!04\)\( p^{8} T^{10} + 8158394864313340 p^{16} T^{12} - 133180568 p^{24} T^{14} + p^{32} T^{16} \) |
| 71 | \( 1 - 114900728 T^{2} + 5630532694361020 T^{4} - \)\(16\!\cdots\!84\)\( T^{6} + \)\(41\!\cdots\!98\)\( T^{8} - \)\(16\!\cdots\!84\)\( p^{8} T^{10} + 5630532694361020 p^{16} T^{12} - 114900728 p^{24} T^{14} + p^{32} T^{16} \) |
| 73 | \( ( 1 - 10620 T + 117578932 T^{2} - 858789498420 T^{3} + 5088110834560998 T^{4} - 858789498420 p^{4} T^{5} + 117578932 p^{8} T^{6} - 10620 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 79 | \( 1 - 140255192 T^{2} + 8310557786451196 T^{4} - \)\(27\!\cdots\!92\)\( T^{6} + \)\(80\!\cdots\!94\)\( T^{8} - \)\(27\!\cdots\!92\)\( p^{8} T^{10} + 8310557786451196 p^{16} T^{12} - 140255192 p^{24} T^{14} + p^{32} T^{16} \) |
| 83 | \( 1 - 233757896 T^{2} + 27717493814314780 T^{4} - \)\(21\!\cdots\!84\)\( T^{6} + \)\(12\!\cdots\!34\)\( T^{8} - \)\(21\!\cdots\!84\)\( p^{8} T^{10} + 27717493814314780 p^{16} T^{12} - 233757896 p^{24} T^{14} + p^{32} T^{16} \) |
| 89 | \( ( 1 - 4584 T + 214096700 T^{2} - 841974282456 T^{3} + 19021468988012934 T^{4} - 841974282456 p^{4} T^{5} + 214096700 p^{8} T^{6} - 4584 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 97 | \( ( 1 - 3480 T + 131157316 T^{2} + 1281368598840 T^{3} + 1798051393872006 T^{4} + 1281368598840 p^{4} T^{5} + 131157316 p^{8} T^{6} - 3480 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
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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
−3.59930471527087595342539428548, −3.41587941051083991725544327512, −3.31295656423894418521783537449, −3.10764803548257806400511671900, −2.84310099749934023508927199378, −2.83787371197161824027983055652, −2.82127989773116375832834054550, −2.76596384037407801089103237717, −2.51316864117644640852956706744, −2.37425318616262372980589635471, −2.06324036081350789774465288693, −2.01383326497270532963181274949, −1.97493461096402645522146693721, −1.83007297899620819648706004470, −1.80612878321253839574529286784, −1.66452355909040449872041721045, −1.62334943189334001684729645046, −0.76926221515096773956350304735, −0.68034157397706934559221925135, −0.66525198921412407177427161852, −0.65528212612508747821076407852, −0.60549953740933432316062449584, −0.60476687100192789044422404363, −0.55088848593752131731558916701, −0.52439471162102541857027289998,
0.52439471162102541857027289998, 0.55088848593752131731558916701, 0.60476687100192789044422404363, 0.60549953740933432316062449584, 0.65528212612508747821076407852, 0.66525198921412407177427161852, 0.68034157397706934559221925135, 0.76926221515096773956350304735, 1.62334943189334001684729645046, 1.66452355909040449872041721045, 1.80612878321253839574529286784, 1.83007297899620819648706004470, 1.97493461096402645522146693721, 2.01383326497270532963181274949, 2.06324036081350789774465288693, 2.37425318616262372980589635471, 2.51316864117644640852956706744, 2.76596384037407801089103237717, 2.82127989773116375832834054550, 2.83787371197161824027983055652, 2.84310099749934023508927199378, 3.10764803548257806400511671900, 3.31295656423894418521783537449, 3.41587941051083991725544327512, 3.59930471527087595342539428548
Plot not available for L-functions of degree greater than 10.
