| L(s) = 1 | + 8·4-s + 8·11-s + 8·13-s + 28·16-s + 24·29-s − 32·31-s + 16·37-s + 16·41-s + 64·44-s − 16·47-s + 64·52-s − 32·61-s + 48·64-s + 16·67-s − 24·71-s + 32·73-s + 16·79-s − 2·81-s − 32·89-s − 16·97-s − 56·103-s − 16·107-s − 32·109-s − 24·113-s + 192·116-s + 32·121-s − 256·124-s + ⋯ |
| L(s) = 1 | + 4·4-s + 2.41·11-s + 2.21·13-s + 7·16-s + 4.45·29-s − 5.74·31-s + 2.63·37-s + 2.49·41-s + 9.64·44-s − 2.33·47-s + 8.87·52-s − 4.09·61-s + 6·64-s + 1.95·67-s − 2.84·71-s + 3.74·73-s + 1.80·79-s − 2/9·81-s − 3.39·89-s − 1.62·97-s − 5.51·103-s − 1.54·107-s − 3.06·109-s − 2.25·113-s + 17.8·116-s + 2.90·121-s − 22.9·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.578052483\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.578052483\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 + T^{4} )^{2} \) |
| 17 | \( 1 \) |
| good | 2 | \( ( 1 - p^{2} T^{2} + 5 p T^{4} - p^{4} T^{6} + p^{4} T^{8} )^{2} \) |
| 5 | \( 1 - 16 T^{3} - 2 T^{4} + 16 p T^{5} + 128 T^{6} + 16 T^{7} - 1149 T^{8} + 16 p T^{9} + 128 p^{2} T^{10} + 16 p^{4} T^{11} - 2 p^{4} T^{12} - 16 p^{5} T^{13} + p^{8} T^{16} \) |
| 7 | \( ( 1 - 8 T + 16 T^{2} + 8 p T^{3} - 320 T^{4} + 8 p^{2} T^{5} + 16 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )( 1 + 8 T + 48 T^{2} + 184 T^{3} + 576 T^{4} + 184 p T^{5} + 48 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} ) \) |
| 11 | \( 1 - 8 T + 32 T^{2} - 112 T^{3} + 574 T^{4} - 2792 T^{5} + 10240 T^{6} - 33960 T^{7} + 111651 T^{8} - 33960 p T^{9} + 10240 p^{2} T^{10} - 2792 p^{3} T^{11} + 574 p^{4} T^{12} - 112 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 13 | \( ( 1 - 4 T + 38 T^{2} - 88 T^{3} + 603 T^{4} - 88 p T^{5} + 38 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 19 | \( 1 - 44 T^{2} + 1514 T^{4} - 39056 T^{6} + 868659 T^{8} - 39056 p^{2} T^{10} + 1514 p^{4} T^{12} - 44 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( 1 - 336 T^{3} + 686 T^{4} + 3696 T^{5} + 56448 T^{6} - 183792 T^{7} - 608141 T^{8} - 183792 p T^{9} + 56448 p^{2} T^{10} + 3696 p^{3} T^{11} + 686 p^{4} T^{12} - 336 p^{5} T^{13} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 12 T + 72 T^{2} - 516 T^{3} + 3502 T^{4} - 516 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( 1 + 32 T + 512 T^{2} + 5824 T^{3} + 55380 T^{4} + 461824 T^{5} + 3383296 T^{6} + 21973472 T^{7} + 128503142 T^{8} + 21973472 p T^{9} + 3383296 p^{2} T^{10} + 461824 p^{3} T^{11} + 55380 p^{4} T^{12} + 5824 p^{5} T^{13} + 512 p^{6} T^{14} + 32 p^{7} T^{15} + p^{8} T^{16} \) |
| 37 | \( 1 - 16 T + 128 T^{2} - 576 T^{3} + 704 T^{4} - 1984 T^{5} + 107520 T^{6} - 1418320 T^{7} + 11216226 T^{8} - 1418320 p T^{9} + 107520 p^{2} T^{10} - 1984 p^{3} T^{11} + 704 p^{4} T^{12} - 576 p^{5} T^{13} + 128 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) |
| 41 | \( 1 - 16 T + 128 T^{2} - 496 T^{3} + 5182 T^{4} - 74112 T^{5} + 384 p^{2} T^{6} - 67616 p T^{7} + 11795619 T^{8} - 67616 p^{2} T^{9} + 384 p^{4} T^{10} - 74112 p^{3} T^{11} + 5182 p^{4} T^{12} - 496 p^{5} T^{13} + 128 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) |
| 43 | \( 1 - 4 p T^{2} + 17642 T^{4} - 1203536 T^{6} + 60458995 T^{8} - 1203536 p^{2} T^{10} + 17642 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16} \) |
| 47 | \( ( 1 + 8 T + 116 T^{2} + 520 T^{3} + 122 p T^{4} + 520 p T^{5} + 116 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 53 | \( 1 - 248 T^{2} + 29372 T^{4} - 2258056 T^{6} + 132650086 T^{8} - 2258056 p^{2} T^{10} + 29372 p^{4} T^{12} - 248 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( 1 - 288 T^{2} + 42592 T^{4} - 4146848 T^{6} + 287130338 T^{8} - 4146848 p^{2} T^{10} + 42592 p^{4} T^{12} - 288 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( 1 + 32 T + 512 T^{2} + 5552 T^{3} + 51392 T^{4} + 472944 T^{5} + 4233856 T^{6} + 34437152 T^{7} + 267409954 T^{8} + 34437152 p T^{9} + 4233856 p^{2} T^{10} + 472944 p^{3} T^{11} + 51392 p^{4} T^{12} + 5552 p^{5} T^{13} + 512 p^{6} T^{14} + 32 p^{7} T^{15} + p^{8} T^{16} \) |
| 67 | \( ( 1 - 8 T + 272 T^{2} - 1592 T^{3} + 27474 T^{4} - 1592 p T^{5} + 272 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( 1 + 24 T + 288 T^{2} + 2616 T^{3} + 26108 T^{4} + 294744 T^{5} + 2976480 T^{6} + 25958328 T^{7} + 218272966 T^{8} + 25958328 p T^{9} + 2976480 p^{2} T^{10} + 294744 p^{3} T^{11} + 26108 p^{4} T^{12} + 2616 p^{5} T^{13} + 288 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) |
| 73 | \( 1 - 32 T + 512 T^{2} - 4448 T^{3} + 13380 T^{4} + 130144 T^{5} - 1122816 T^{6} - 6615264 T^{7} + 145221574 T^{8} - 6615264 p T^{9} - 1122816 p^{2} T^{10} + 130144 p^{3} T^{11} + 13380 p^{4} T^{12} - 4448 p^{5} T^{13} + 512 p^{6} T^{14} - 32 p^{7} T^{15} + p^{8} T^{16} \) |
| 79 | \( 1 - 16 T + 128 T^{2} - 1616 T^{3} + 29024 T^{4} - 275888 T^{5} + 2004864 T^{6} - 23688432 T^{7} + 277955074 T^{8} - 23688432 p T^{9} + 2004864 p^{2} T^{10} - 275888 p^{3} T^{11} + 29024 p^{4} T^{12} - 1616 p^{5} T^{13} + 128 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) |
| 83 | \( 1 - 360 T^{2} + 62012 T^{4} - 7308440 T^{6} + 675643686 T^{8} - 7308440 p^{2} T^{10} + 62012 p^{4} T^{12} - 360 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 16 T + 336 T^{2} + 3376 T^{3} + 41408 T^{4} + 3376 p T^{5} + 336 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 + 16 T + 128 T^{2} + 1680 T^{3} + 30164 T^{4} + 287792 T^{5} + 2154880 T^{6} + 25520304 T^{7} + 301708390 T^{8} + 25520304 p T^{9} + 2154880 p^{2} T^{10} + 287792 p^{3} T^{11} + 30164 p^{4} T^{12} + 1680 p^{5} T^{13} + 128 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) |
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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
−4.34093364322214796285205971976, −4.28397848126314125784337044078, −4.02431605871554414866357927375, −3.86041690952362724884714435460, −3.83870682825965808457179666801, −3.65343914426058154906818322499, −3.62362928261277679685482522496, −3.42976209005308353805700709677, −3.06706783832225280189853984728, −3.05142305579154096312692716348, −3.04065906785246803659516070751, −2.79809676191800663281180351230, −2.63421034856324729032930441372, −2.54301662188662547141164053235, −2.42841130222436896292885273416, −2.22022215621238628556517253084, −2.03132138855613656369072565234, −1.91230724956300287767105876537, −1.64833278688601214016250950196, −1.41937273512072726261388062983, −1.27654562213331550814439556268, −1.23098303480191781136738917065, −1.22189926938683465050434602663, −0.998580179186258065539660685338, −0.07248925801187343866692222835,
0.07248925801187343866692222835, 0.998580179186258065539660685338, 1.22189926938683465050434602663, 1.23098303480191781136738917065, 1.27654562213331550814439556268, 1.41937273512072726261388062983, 1.64833278688601214016250950196, 1.91230724956300287767105876537, 2.03132138855613656369072565234, 2.22022215621238628556517253084, 2.42841130222436896292885273416, 2.54301662188662547141164053235, 2.63421034856324729032930441372, 2.79809676191800663281180351230, 3.04065906785246803659516070751, 3.05142305579154096312692716348, 3.06706783832225280189853984728, 3.42976209005308353805700709677, 3.62362928261277679685482522496, 3.65343914426058154906818322499, 3.83870682825965808457179666801, 3.86041690952362724884714435460, 4.02431605871554414866357927375, 4.28397848126314125784337044078, 4.34093364322214796285205971976
Plot not available for L-functions of degree greater than 10.
