| L(s) = 1 | − 532·9-s − 352·11-s + 1.96e3·23-s − 5.78e3·25-s + 4.04e3·29-s + 1.05e4·37-s + 2.87e4·43-s − 2.82e4·53-s − 7.25e4·67-s + 1.76e5·71-s + 1.35e5·79-s + 1.40e5·81-s + 1.87e5·99-s + 2.27e5·107-s − 7.65e4·109-s + 4.71e5·113-s + 2.62e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 6.24e5·169-s + ⋯ |
| L(s) = 1 | − 2.18·9-s − 0.877·11-s + 0.775·23-s − 1.85·25-s + 0.892·29-s + 1.26·37-s + 2.37·43-s − 1.38·53-s − 1.97·67-s + 4.16·71-s + 2.45·79-s + 2.37·81-s + 1.92·99-s + 1.92·107-s − 0.616·109-s + 3.47·113-s + 1.63·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s − 1.68·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.386917525\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.386917525\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 532 T^{2} + 15862 p^{2} T^{4} + 532 p^{10} T^{6} + p^{20} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 + 5788 T^{2} + 17534886 T^{4} + 5788 p^{10} T^{6} + p^{20} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 16 p T - 85018 T^{2} + 16 p^{6} T^{3} + p^{10} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 624444 T^{2} + 200605765382 T^{4} + 624444 p^{10} T^{6} + p^{20} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 2118244 T^{2} + 3693664806918 T^{4} + 2118244 p^{10} T^{6} + p^{20} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 2402388 T^{2} + 8781383217638 T^{4} + 2402388 p^{10} T^{6} + p^{20} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 984 T + 9380974 T^{2} - 984 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 1010 T + p^{5} T^{2} )^{4} \) |
| 31 | $D_4\times C_2$ | \( 1 - 26189412 T^{2} + 734880264661574 T^{4} - 26189412 p^{10} T^{6} + p^{20} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 5252 T + 130648686 T^{2} - 5252 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 69556804 T^{2} + 26464847907081510 T^{4} + 69556804 p^{10} T^{6} + p^{20} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 14368 T + 312022758 T^{2} - 14368 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 738897628 T^{2} + 239552681984211078 T^{4} + 738897628 p^{10} T^{6} + p^{20} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 14148 T + 287368846 T^{2} + 14148 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 1222868596 T^{2} + 1280745363811545606 T^{4} + 1222868596 p^{10} T^{6} + p^{20} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 1878217404 T^{2} + 2308173235702714310 T^{4} + 1878217404 p^{10} T^{6} + p^{20} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 36288 T + 2332068566 T^{2} + 36288 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 88368 T + 4828864462 T^{2} - 88368 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 1139195812 T^{2} - 3263056635172107162 T^{4} + 1139195812 p^{10} T^{6} + p^{20} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 67984 T + 7283017566 T^{2} - 67984 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 2940509420 T^{2} + 26967102015558640998 T^{4} - 2940509420 p^{10} T^{6} + p^{20} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 16950702308 T^{2} + \)\(12\!\cdots\!18\)\( T^{4} + 16950702308 p^{10} T^{6} + p^{20} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 33577194660 T^{2} + \)\(42\!\cdots\!98\)\( T^{4} + 33577194660 p^{10} T^{6} + p^{20} T^{8} \) |
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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
−7.48422022165212836624681896525, −7.12155520366449994887234434094, −6.87827239588533040743933073707, −6.40536134411456993809138418961, −6.27977087509267962451461122629, −5.91991498555758961013563250021, −5.82039443118259062882382145822, −5.75656244631768410063948962168, −5.37469547776589798879834129774, −4.93332680316589753394450407360, −4.72415042423868416203707774707, −4.62864373480211105975236829927, −4.25749424657934573528990371296, −3.77169321002369577594137225823, −3.37632872950105936188956091948, −3.26599083783188950275749377197, −3.14160497177545290700027486052, −2.49942470610534284340073733498, −2.28585054906805073099770471500, −2.25052866396262659378103929504, −1.86721910853683448370682628431, −1.09112403958573471337710849142, −0.834245591889282431158274504839, −0.54038328486896900084128017308, −0.25085236517063360918497978712,
0.25085236517063360918497978712, 0.54038328486896900084128017308, 0.834245591889282431158274504839, 1.09112403958573471337710849142, 1.86721910853683448370682628431, 2.25052866396262659378103929504, 2.28585054906805073099770471500, 2.49942470610534284340073733498, 3.14160497177545290700027486052, 3.26599083783188950275749377197, 3.37632872950105936188956091948, 3.77169321002369577594137225823, 4.25749424657934573528990371296, 4.62864373480211105975236829927, 4.72415042423868416203707774707, 4.93332680316589753394450407360, 5.37469547776589798879834129774, 5.75656244631768410063948962168, 5.82039443118259062882382145822, 5.91991498555758961013563250021, 6.27977087509267962451461122629, 6.40536134411456993809138418961, 6.87827239588533040743933073707, 7.12155520366449994887234434094, 7.48422022165212836624681896525
