| L(s) = 1 | − 46·9-s − 120·11-s − 4.18e3·19-s − 7.10e3·29-s − 1.77e4·31-s − 2.48e4·41-s + 3.69e4·49-s + 7.33e4·59-s + 3.84e4·61-s + 5.47e3·71-s + 3.23e4·79-s + 4.70e4·81-s + 9.46e4·89-s + 5.52e3·99-s + 2.79e5·101-s + 1.93e5·109-s − 4.51e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5.90e5·169-s + ⋯ |
| L(s) = 1 | − 0.189·9-s − 0.299·11-s − 2.65·19-s − 1.56·29-s − 3.32·31-s − 2.31·41-s + 2.20·49-s + 2.74·59-s + 1.32·61-s + 0.128·71-s + 0.583·79-s + 0.797·81-s + 1.26·89-s + 0.0566·99-s + 2.72·101-s + 1.56·109-s − 2.80·121-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 + 1.59·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.921638460\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.921638460\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 46 T^{2} - 4997 p^{2} T^{4} + 46 p^{10} T^{6} + p^{20} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 36980 T^{2} + 883272198 T^{4} - 36980 p^{10} T^{6} + p^{20} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 60 T + 230977 T^{2} + 60 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 590804 T^{2} + 163581027702 T^{4} - 590804 p^{10} T^{6} + p^{20} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 1327586 T^{2} + 3973871730147 T^{4} - 1327586 p^{10} T^{6} + p^{20} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 2092 T + 5218089 T^{2} + 2092 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 10160180 T^{2} + 108548175292998 T^{4} - 10160180 p^{10} T^{6} + p^{20} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 3552 T + 20618074 T^{2} + 3552 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 8888 T + 67804938 T^{2} + 8888 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 33594380 T^{2} - 2634705186058602 T^{4} - 33594380 p^{10} T^{6} + p^{20} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 12438 T + 217381963 T^{2} + 12438 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 540244172 T^{2} + 116157205788519894 T^{4} - 540244172 p^{10} T^{6} + p^{20} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 902407196 T^{2} + 308772689280765702 T^{4} - 902407196 p^{10} T^{6} + p^{20} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 1189716620 T^{2} + 656395125486896598 T^{4} - 1189716620 p^{10} T^{6} + p^{20} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 36696 T + 1234961302 T^{2} - 36696 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 19204 T + 627029406 T^{2} - 19204 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 974988050 T^{2} + 2516736182389071123 T^{4} - 974988050 p^{10} T^{6} + p^{20} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 2736 T + 2006886526 T^{2} - 2736 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 8204978114 T^{2} + 25425197509477462947 T^{4} - 8204978114 p^{10} T^{6} + p^{20} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 16184 T + 6157384362 T^{2} - 16184 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 7195965410 T^{2} + 40258768525685533923 T^{4} - 7195965410 p^{10} T^{6} + p^{20} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 47322 T - 1006825781 T^{2} - 47322 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 28221621500 T^{2} + \)\(34\!\cdots\!98\)\( T^{4} - 28221621500 p^{10} T^{6} + p^{20} T^{8} \) |
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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
−9.171520367520651931897423683153, −9.056763155725172342628754643332, −8.704288867692366983411201254529, −8.525050751795687472172398540991, −8.020278023549258325100656645952, −7.65839566537453760585535418911, −7.60140058978426840197277078159, −7.11622645047245244263854024629, −6.78814918059449269178103679682, −6.41712478816359752938069374550, −6.40946962161651506076203273307, −5.66923230659832695074357768719, −5.47470339110188845388948768007, −5.11937304874685490671292761062, −5.08369290171369049130081602157, −4.14547754083475205455185677560, −3.93584778520455956939183358130, −3.83335633927490946373800494837, −3.38239138939104770718485434947, −2.67182317019957128939788502375, −2.07406607159161549792539733844, −2.06882364755979776999847731542, −1.61442456900992045042444151405, −0.50644913354941925684782496849, −0.40150117790614352559686119402,
0.40150117790614352559686119402, 0.50644913354941925684782496849, 1.61442456900992045042444151405, 2.06882364755979776999847731542, 2.07406607159161549792539733844, 2.67182317019957128939788502375, 3.38239138939104770718485434947, 3.83335633927490946373800494837, 3.93584778520455956939183358130, 4.14547754083475205455185677560, 5.08369290171369049130081602157, 5.11937304874685490671292761062, 5.47470339110188845388948768007, 5.66923230659832695074357768719, 6.40946962161651506076203273307, 6.41712478816359752938069374550, 6.78814918059449269178103679682, 7.11622645047245244263854024629, 7.60140058978426840197277078159, 7.65839566537453760585535418911, 8.020278023549258325100656645952, 8.525050751795687472172398540991, 8.704288867692366983411201254529, 9.056763155725172342628754643332, 9.171520367520651931897423683153
