| L(s) = 1 | + 13·4-s − 204·13-s + 122·16-s + 222·19-s + 67·25-s + 220·31-s + 374·37-s − 1.67e3·43-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 − 6.02e3·97-s + 871·100-s + 2.04e3·103-s + 1.01e3·109-s + 1.38e3·121-s + 2.86e3·124-s + 127-s + 131-s + 137-s + 139-s + 4.86e3·148-s + 149-s + ⋯ |
| L(s) = 1 | + 13/8·4-s − 4.35·13-s + 1.90·16-s + 2.68·19-s + 0.535·25-s + 1.27·31-s + 1.66·37-s − 5.94·43-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 − 6.30·97-s + 0.870·100-s + 1.95·103-s + 0.887·109-s + 1.04·121-s + 2.07·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 2.70·148-s + 0.000549·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8520667139\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8520667139\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 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
−4.46130123303381068859719101752, −4.39480154278735111753150064644, −4.23813114195689700267218918145, −4.17982925194599712343988613688, −3.62612067624550848179018250946, −3.51936430320641717766519310541, −3.40141827480821169123382613892, −3.37615927301197591612508589601, −3.15265786417673857755342412179, −3.03088797466165391203133167119, −2.83607988444373655196546559093, −2.67206744077918775420891361099, −2.63801065460833442558346961846, −2.50415651788798355253027539806, −2.17535039989789884951937691489, −2.10202303477560897939539403830, −1.90740260159522533090452093968, −1.67496871989813213219340009395, −1.49412159310382378800351305337, −1.35965778053180748168890093209, −1.12056010431568621489085166333, −0.70813922531276922751971213625, −0.64054517918617182664615303989, −0.43395283997814735699514128858, −0.05924886250288746482302525167,
0.05924886250288746482302525167, 0.43395283997814735699514128858, 0.64054517918617182664615303989, 0.70813922531276922751971213625, 1.12056010431568621489085166333, 1.35965778053180748168890093209, 1.49412159310382378800351305337, 1.67496871989813213219340009395, 1.90740260159522533090452093968, 2.10202303477560897939539403830, 2.17535039989789884951937691489, 2.50415651788798355253027539806, 2.63801065460833442558346961846, 2.67206744077918775420891361099, 2.83607988444373655196546559093, 3.03088797466165391203133167119, 3.15265786417673857755342412179, 3.37615927301197591612508589601, 3.40141827480821169123382613892, 3.51936430320641717766519310541, 3.62612067624550848179018250946, 4.17982925194599712343988613688, 4.23813114195689700267218918145, 4.39480154278735111753150064644, 4.46130123303381068859719101752
Plot not available for L-functions of degree greater than 10.
