| L(s) = 1 | + 4·5-s − 6·7-s + 52·11-s + 26·13-s + 188·17-s − 74·19-s + 148·23-s − 58·25-s + 288·29-s − 248·31-s − 24·35-s + 342·37-s − 256·43-s + 132·47-s + 61·49-s + 952·53-s + 208·55-s − 1.00e3·59-s − 34·61-s + 104·65-s − 866·67-s + 776·71-s + 1.87e3·73-s − 312·77-s + 182·79-s − 1.33e3·83-s + 752·85-s + ⋯ |
| L(s) = 1 | + 0.357·5-s − 0.323·7-s + 1.42·11-s + 0.554·13-s + 2.68·17-s − 0.893·19-s + 1.34·23-s − 0.463·25-s + 1.84·29-s − 1.43·31-s − 0.115·35-s + 1.51·37-s − 0.907·43-s + 0.409·47-s + 0.177·49-s + 2.46·53-s + 0.509·55-s − 2.21·59-s − 0.0713·61-s + 0.198·65-s − 1.57·67-s + 1.29·71-s + 3.00·73-s − 0.461·77-s + 0.259·79-s − 1.76·83-s + 0.959·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.382458558\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.382458558\) |
| \(L(\frac{5}{2})\) |
|
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 | $D_{4}$ | \( 1 - 4 T + 74 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 6 T - 25 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 52 T + 1718 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 p T + 3843 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 188 T + 18482 T^{2} - 188 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 74 T + 3567 T^{2} + 74 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 148 T + 25310 T^{2} - 148 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 288 T + 57994 T^{2} - 288 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 p T + 63438 T^{2} + 8 p^{4} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 342 T + 112547 T^{2} - 342 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 46478 T^{2} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 256 T + 172518 T^{2} + 256 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 132 T + 211822 T^{2} - 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 952 T + 489050 T^{2} - 952 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 1004 T + 583382 T^{2} + 1004 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 34 T + 246171 T^{2} + 34 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 866 T + 742935 T^{2} + 866 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 776 T + 704366 T^{2} - 776 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1874 T + 1630083 T^{2} - 1874 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 182 T + 872679 T^{2} - 182 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1336 T + 1208918 T^{2} + 1336 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 876 T + 1522402 T^{2} - 876 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 38 T + 431787 T^{2} + 38 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.01477360564721692149909106412, −11.87189718680763448612305672903, −11.04830533739975083452403542113, −10.68428844058151647109331366138, −10.10034737706200748965585349881, −9.700019500606392757199705238183, −9.109963385957755448638485621114, −8.874460841792595789490260716579, −7.994984073680498576164950890446, −7.76006998097484215205936167388, −6.76074817815947818101314363036, −6.64402041529322826585320920279, −5.82010827466793141936687172210, −5.50572886747401703011439387009, −4.64736962802747670093529492348, −3.85807568515177784708004004153, −3.40688375066134553510120077105, −2.60916989393121595807596145526, −1.40509757227585239736566932328, −0.904222850523817748146116873196,
0.904222850523817748146116873196, 1.40509757227585239736566932328, 2.60916989393121595807596145526, 3.40688375066134553510120077105, 3.85807568515177784708004004153, 4.64736962802747670093529492348, 5.50572886747401703011439387009, 5.82010827466793141936687172210, 6.64402041529322826585320920279, 6.76074817815947818101314363036, 7.76006998097484215205936167388, 7.994984073680498576164950890446, 8.874460841792595789490260716579, 9.109963385957755448638485621114, 9.700019500606392757199705238183, 10.10034737706200748965585349881, 10.68428844058151647109331366138, 11.04830533739975083452403542113, 11.87189718680763448612305672903, 12.01477360564721692149909106412
