| L(s) = 1 | + 12·3-s − 60·7-s + 90·9-s − 720·21-s − 400·23-s + 540·27-s − 84·29-s + 1.24e3·41-s − 192·43-s + 440·47-s + 1.53e3·49-s + 552·61-s − 5.40e3·63-s + 408·67-s − 4.80e3·69-s + 2.83e3·81-s + 608·83-s − 1.00e3·87-s + 312·89-s − 1.30e3·101-s − 396·103-s + 3.71e3·107-s + 1.53e3·109-s + 1.76e3·121-s + 1.49e4·123-s + 127-s − 2.30e3·129-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 3.23·7-s + 10/3·9-s − 7.48·21-s − 3.62·23-s + 3.84·27-s − 0.537·29-s + 4.75·41-s − 0.680·43-s + 1.36·47-s + 4.47·49-s + 1.15·61-s − 10.7·63-s + 0.743·67-s − 8.37·69-s + 35/9·81-s + 0.804·83-s − 1.24·87-s + 0.371·89-s − 1.28·101-s − 0.378·103-s + 3.35·107-s + 1.34·109-s + 1.32·121-s + 10.9·123-s + 0.000698·127-s − 1.57·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.948739065\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.948739065\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( ( 1 + 30 T + 582 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 1768 T^{2} + 4028478 T^{4} - 1768 p^{6} T^{6} + p^{12} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 3632 T^{2} + 56382 p^{2} T^{4} - 3632 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 12272 T^{2} + 83923998 T^{4} + 12272 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 6124 T^{2} + 27666006 T^{4} + 6124 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $C_2$ | \( ( 1 + 100 T + p^{3} T^{2} )^{4} \) |
| 29 | $D_{4}$ | \( ( 1 + 42 T + 46258 T^{2} + 42 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 59692 T^{2} + 2433648678 T^{4} + 59692 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 28336 T^{2} - 2229036402 T^{4} + 28336 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 624 T + 202286 T^{2} - 624 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 96 T + 29718 T^{2} + 96 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 - 220 T + 113150 T^{2} - 220 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 501872 T^{2} + 106308375630 T^{4} + 501872 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 802616 T^{2} + 245338955070 T^{4} + 802616 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 276 T + 283502 T^{2} - 276 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 204 T + 452694 T^{2} - 204 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 + 1289228 T^{2} + 671568403974 T^{4} + 1289228 p^{6} T^{6} + p^{12} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 4468 p T^{2} + 327406862598 T^{4} + 4468 p^{7} T^{6} + p^{12} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 1912684 T^{2} + 1400532789606 T^{4} + 1912684 p^{6} T^{6} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 304 T + 1119302 T^{2} - 304 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 156 T + 1226518 T^{2} - 156 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 2557444 T^{2} + 3281668753542 T^{4} + 2557444 p^{6} T^{6} + p^{12} T^{8} \) |
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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
−6.05900906256459382286037101235, −5.83111042970127777990547085930, −5.70632498332185734112062459029, −5.42815888402120580072328438757, −5.42331091511611551379511275714, −4.65755252255337164925469733565, −4.48803753607357828958547334467, −4.41126378682319557415101937225, −4.23226841207793358487634758069, −3.81306908840915697769578195516, −3.79066924997501301575775137907, −3.58217253574633856493256609173, −3.51855653513103167264916098246, −3.16331733616599652013292737268, −2.76235150769157128298282281068, −2.72144338177831433497423338443, −2.69939928762803291850941093247, −2.22716850560531748124466154988, −1.96949820855539155461143068415, −1.86203513919760105019412870772, −1.70595193877541310289151756001, −0.77064667741259562261149205442, −0.76295764350744276908756855379, −0.67477764011004420021762333221, −0.18137073427853652610029384846,
0.18137073427853652610029384846, 0.67477764011004420021762333221, 0.76295764350744276908756855379, 0.77064667741259562261149205442, 1.70595193877541310289151756001, 1.86203513919760105019412870772, 1.96949820855539155461143068415, 2.22716850560531748124466154988, 2.69939928762803291850941093247, 2.72144338177831433497423338443, 2.76235150769157128298282281068, 3.16331733616599652013292737268, 3.51855653513103167264916098246, 3.58217253574633856493256609173, 3.79066924997501301575775137907, 3.81306908840915697769578195516, 4.23226841207793358487634758069, 4.41126378682319557415101937225, 4.48803753607357828958547334467, 4.65755252255337164925469733565, 5.42331091511611551379511275714, 5.42815888402120580072328438757, 5.70632498332185734112062459029, 5.83111042970127777990547085930, 6.05900906256459382286037101235
