| L(s) = 1 | + 4·5-s + 6·7-s + 10·11-s + 40·13-s + 68·17-s + 68·19-s − 98·23-s + 125·25-s − 120·29-s + 200·31-s + 24·35-s + 960·37-s − 192·41-s + 334·43-s − 300·47-s + 566·49-s + 136·53-s + 40·55-s − 620·59-s − 200·61-s + 160·65-s + 1.40e3·67-s + 2.78e3·71-s + 3.60e3·73-s + 60·77-s + 334·79-s + 500·83-s + ⋯ |
| L(s) = 1 | + 0.357·5-s + 0.323·7-s + 0.274·11-s + 0.853·13-s + 0.970·17-s + 0.821·19-s − 0.888·23-s + 25-s − 0.768·29-s + 1.15·31-s + 0.115·35-s + 4.26·37-s − 0.731·41-s + 1.18·43-s − 0.931·47-s + 1.65·49-s + 0.352·53-s + 0.0980·55-s − 1.36·59-s − 0.419·61-s + 0.305·65-s + 2.56·67-s + 4.64·71-s + 5.77·73-s + 0.0888·77-s + 0.475·79-s + 0.661·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(14.96075168\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.96075168\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 4 T - 109 T^{2} + 4 p^{3} T^{3} - 16 p^{3} T^{4} + 4 p^{6} T^{5} - 109 p^{6} T^{6} - 4 p^{9} T^{7} + p^{12} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 6 T - 530 T^{2} + 720 T^{3} + 190359 T^{4} + 720 p^{3} T^{5} - 530 p^{6} T^{6} - 6 p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 10 T - 1426 T^{2} + 11360 T^{3} + 424015 T^{4} + 11360 p^{3} T^{5} - 1426 p^{6} T^{6} - 10 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 40 T + 31 T^{2} + 113000 T^{3} - 5880248 T^{4} + 113000 p^{3} T^{5} + 31 p^{6} T^{6} - 40 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 2 p T + 8051 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 34 T + 10782 T^{2} - 34 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 98 T - 470 p T^{2} - 384160 T^{3} + 151843639 T^{4} - 384160 p^{3} T^{5} - 470 p^{7} T^{6} + 98 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 120 T - 22369 T^{2} - 1441080 T^{3} + 405934440 T^{4} - 1441080 p^{3} T^{5} - 22369 p^{6} T^{6} + 120 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 200 T - 21326 T^{2} - 348800 T^{3} + 1681734595 T^{4} - 348800 p^{3} T^{5} - 21326 p^{6} T^{6} - 200 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 480 T + 152585 T^{2} - 480 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 192 T - 97294 T^{2} - 707328 T^{3} + 10707560979 T^{4} - 707328 p^{3} T^{5} - 97294 p^{6} T^{6} + 192 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 334 T + 5278 T^{2} + 17613824 T^{3} - 3895832657 T^{4} + 17613824 p^{3} T^{5} + 5278 p^{6} T^{6} - 334 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 300 T + 46130 T^{2} - 49132800 T^{3} - 18198708429 T^{4} - 49132800 p^{3} T^{5} + 46130 p^{6} T^{6} + 300 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 68 T + 166814 T^{2} - 68 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 620 T - 117814 T^{2} + 56702720 T^{3} + 131090680555 T^{4} + 56702720 p^{3} T^{5} - 117814 p^{6} T^{6} + 620 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 200 T - 265937 T^{2} - 29605000 T^{3} + 32997833608 T^{4} - 29605000 p^{3} T^{5} - 265937 p^{6} T^{6} + 200 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 1406 T + 896710 T^{2} - 672911600 T^{3} + 481654667839 T^{4} - 672911600 p^{3} T^{5} + 896710 p^{6} T^{6} - 1406 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 1390 T + 1197686 T^{2} - 1390 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 1802 T + 1548039 T^{2} - 1802 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 334 T - 97322 T^{2} + 259584800 T^{3} - 254458895321 T^{4} + 259584800 p^{3} T^{5} - 97322 p^{6} T^{6} - 334 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 500 T - 249670 T^{2} + 321952000 T^{3} - 220217014469 T^{4} + 321952000 p^{3} T^{5} - 249670 p^{6} T^{6} - 500 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 90 T + 1337659 T^{2} - 90 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 200 T + 124690 T^{2} + 382007200 T^{3} - 862306528829 T^{4} + 382007200 p^{3} T^{5} + 124690 p^{6} T^{6} - 200 p^{9} T^{7} + p^{12} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.41326096970394010471465463078, −6.75260639244918335900350190077, −6.66711458413354949037089874890, −6.35876032297143470079720294631, −6.26808375427947758207567277258, −6.14698471684070785360939733270, −5.65173640235751729019635367398, −5.51184055342439213003453943573, −5.40020225263797479966968534966, −4.85402608416621801770778806442, −4.77041989462041670949314571297, −4.47075793541465010321415249741, −4.39975671625258804590616839417, −3.59704686990743989085639066156, −3.57191945024976938432361938087, −3.45614152052246735214718721968, −3.42052764873272440126137449703, −2.37774979762269198018557754312, −2.36544467832080442218577852656, −2.33529627726776542476107029889, −1.96839526764580073171407756559, −1.05050541809741534273719374652, −0.993414654760538544600297941068, −0.833888039066121189104982469421, −0.59084445526429694786433079182,
0.59084445526429694786433079182, 0.833888039066121189104982469421, 0.993414654760538544600297941068, 1.05050541809741534273719374652, 1.96839526764580073171407756559, 2.33529627726776542476107029889, 2.36544467832080442218577852656, 2.37774979762269198018557754312, 3.42052764873272440126137449703, 3.45614152052246735214718721968, 3.57191945024976938432361938087, 3.59704686990743989085639066156, 4.39975671625258804590616839417, 4.47075793541465010321415249741, 4.77041989462041670949314571297, 4.85402608416621801770778806442, 5.40020225263797479966968534966, 5.51184055342439213003453943573, 5.65173640235751729019635367398, 6.14698471684070785360939733270, 6.26808375427947758207567277258, 6.35876032297143470079720294631, 6.66711458413354949037089874890, 6.75260639244918335900350190077, 7.41326096970394010471465463078
