| L(s) = 1 | + 13·4-s − 12·7-s + 204·13-s + 122·16-s − 222·19-s + 67·25-s − 156·28-s − 220·31-s + 374·37-s − 1.67e3·43-s + 262·49-s + 2.65e3·52-s − 1.33e3·61-s + 897·64-s − 1.89e3·67-s − 1.75e3·73-s − 2.88e3·76-s − 8·79-s − 2.44e3·91-s + 6.02e3·97-s + 871·100-s − 2.04e3·103-s + 1.01e3·109-s − 1.46e3·112-s + 1.38e3·121-s − 2.86e3·124-s + 127-s + ⋯ |
| L(s) = 1 | + 13/8·4-s − 0.647·7-s + 4.35·13-s + 1.90·16-s − 2.68·19-s + 0.535·25-s − 1.05·28-s − 1.27·31-s + 1.66·37-s − 5.94·43-s + 0.763·49-s + 7.07·52-s − 2.79·61-s + 1.75·64-s − 3.44·67-s − 2.80·73-s − 4.35·76-s − 0.0113·79-s − 2.82·91-s + 6.30·97-s + 0.870·100-s − 1.95·103-s + 0.887·109-s − 1.23·112-s + 1.04·121-s − 2.07·124-s + 0.000698·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.697654436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.697654436\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( ( 1 + 6 T - 11 p T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| good | 2 | \( 1 - 13 T^{2} + 47 T^{4} + 39 p T^{6} - 7 p^{2} T^{8} + 39 p^{7} T^{10} + 47 p^{12} T^{12} - 13 p^{18} T^{14} + p^{24} T^{16} \) |
| 5 | \( 1 - 67 T^{2} - 13939 T^{4} + 859074 T^{6} + 25309934 T^{8} + 859074 p^{6} T^{10} - 13939 p^{12} T^{12} - 67 p^{18} T^{14} + p^{24} T^{16} \) |
| 11 | \( 1 - 1387 T^{2} + 98779 p T^{4} + 3753113814 T^{6} - 5845598670130 T^{8} + 3753113814 p^{6} T^{10} + 98779 p^{13} T^{12} - 1387 p^{18} T^{14} + p^{24} T^{16} \) |
| 13 | \( ( 1 - 51 T + 4996 T^{2} - 51 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 17 | \( 1 - 4256 T^{2} - 33515774 T^{4} - 14275356032 T^{6} + 1645746688812355 T^{8} - 14275356032 p^{6} T^{10} - 33515774 p^{12} T^{12} - 4256 p^{18} T^{14} + p^{24} T^{16} \) |
| 19 | \( ( 1 + 111 T + 1361 T^{2} - 306138 T^{3} - 11434020 T^{4} - 306138 p^{3} T^{5} + 1361 p^{6} T^{6} + 111 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 23 | \( 1 - 8216 T^{2} + 118915282 T^{4} + 2854931863264 T^{6} - 29264182010625629 T^{8} + 2854931863264 p^{6} T^{10} + 118915282 p^{12} T^{12} - 8216 p^{18} T^{14} + p^{24} T^{16} \) |
| 29 | \( ( 1 + 59371 T^{2} + 1802370044 T^{4} + 59371 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 31 | \( ( 1 + 110 T - 31207 T^{2} - 57750 p T^{3} + 652988 p^{2} T^{4} - 57750 p^{4} T^{5} - 31207 p^{6} T^{6} + 110 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 37 | \( ( 1 - 187 T - 44923 T^{2} + 4004418 T^{3} + 2045720498 T^{4} + 4004418 p^{3} T^{5} - 44923 p^{6} T^{6} - 187 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 41 | \( ( 1 + 85540 T^{2} + 8435951270 T^{4} + 85540 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 43 | \( ( 1 + 419 T + 167730 T^{2} + 419 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 47 | \( 1 - 279692 T^{2} + 790476854 p T^{4} - 5458685013243056 T^{6} + \)\(72\!\cdots\!79\)\( T^{8} - 5458685013243056 p^{6} T^{10} + 790476854 p^{13} T^{12} - 279692 p^{18} T^{14} + p^{24} T^{16} \) |
| 53 | \( 1 - 574307 T^{2} + 203127896557 T^{4} - 47306765339926238 T^{6} + \)\(82\!\cdots\!02\)\( T^{8} - 47306765339926238 p^{6} T^{10} + 203127896557 p^{12} T^{12} - 574307 p^{18} T^{14} + p^{24} T^{16} \) |
| 59 | \( 1 - 615851 T^{2} + 202986056233 T^{4} - 56612306828303786 T^{6} + \)\(13\!\cdots\!02\)\( T^{8} - 56612306828303786 p^{6} T^{10} + 202986056233 p^{12} T^{12} - 615851 p^{18} T^{14} + p^{24} T^{16} \) |
| 61 | \( ( 1 + 666 T - 19198 T^{2} + 5855472 T^{3} + 61942105719 T^{4} + 5855472 p^{3} T^{5} - 19198 p^{6} T^{6} + 666 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 67 | \( ( 1 + 945 T + 85661 T^{2} + 194516910 T^{3} + 292789368252 T^{4} + 194516910 p^{3} T^{5} + 85661 p^{6} T^{6} + 945 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 71 | \( ( 1 + 874364 T^{2} + 392544633894 T^{4} + 874364 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 73 | \( ( 1 + 875 T - 201451 T^{2} + 165411750 T^{3} + 486060622862 T^{4} + 165411750 p^{3} T^{5} - 201451 p^{6} T^{6} + 875 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 79 | \( ( 1 + 4 T - 900953 T^{2} - 340436 T^{3} + 568649794816 T^{4} - 340436 p^{3} T^{5} - 900953 p^{6} T^{6} + 4 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 83 | \( ( 1 + 1482323 T^{2} + 1051851285456 T^{4} + 1482323 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 89 | \( 1 - 627904 T^{2} - 652647550942 T^{4} - 33246513557593344 T^{6} + \)\(65\!\cdots\!67\)\( T^{8} - 33246513557593344 p^{6} T^{10} - 652647550942 p^{12} T^{12} - 627904 p^{18} T^{14} + p^{24} T^{16} \) |
| 97 | \( ( 1 - 1505 T + 1992044 T^{2} - 1505 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
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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
−6.39273794203873742440701597406, −6.35651953521431817178971895270, −6.27759854454298615848025521097, −5.98582912559248432474082167657, −5.97981765506655478649403186545, −5.94670761193068942015686945801, −5.88428862609934178753440745886, −5.21242257532195816276155255538, −5.13935554047640458773678096032, −4.72268867019042596916604288154, −4.59905531405162958262248200220, −4.47591753034872195676139374521, −4.16079453089427303410431204321, −3.76008857671432544824702939226, −3.51422044183024914902467371682, −3.41440260665569850967033651579, −3.38280915997644845325388825441, −3.12677133772204655938795484185, −2.66407687070997829137026273793, −2.34415855697655614374052881179, −1.92738238983978013877255190590, −1.57294970242384895169638812720, −1.42920393168876694624862623410, −1.22070740853299530561832500641, −0.20086376823137974777784405416,
0.20086376823137974777784405416, 1.22070740853299530561832500641, 1.42920393168876694624862623410, 1.57294970242384895169638812720, 1.92738238983978013877255190590, 2.34415855697655614374052881179, 2.66407687070997829137026273793, 3.12677133772204655938795484185, 3.38280915997644845325388825441, 3.41440260665569850967033651579, 3.51422044183024914902467371682, 3.76008857671432544824702939226, 4.16079453089427303410431204321, 4.47591753034872195676139374521, 4.59905531405162958262248200220, 4.72268867019042596916604288154, 5.13935554047640458773678096032, 5.21242257532195816276155255538, 5.88428862609934178753440745886, 5.94670761193068942015686945801, 5.97981765506655478649403186545, 5.98582912559248432474082167657, 6.27759854454298615848025521097, 6.35651953521431817178971895270, 6.39273794203873742440701597406
Plot not available for L-functions of degree greater than 10.
