| L(s) = 1 | + 28·5-s + 54·7-s + 148·11-s + 242·13-s + 212·17-s − 62·19-s + 2.30e3·23-s − 2.60e3·25-s + 2.40e3·29-s − 968·31-s + 1.51e3·35-s − 8.17e3·37-s + 1.92e4·41-s + 1.12e4·43-s + 2.81e4·47-s − 1.91e4·49-s + 4.69e4·53-s + 4.14e3·55-s + 2.07e4·59-s − 1.17e4·61-s + 6.77e3·65-s + 3.45e4·67-s + 1.03e5·71-s − 5.61e4·73-s + 7.99e3·77-s + 1.60e4·79-s + 8.62e4·83-s + ⋯ |
| L(s) = 1 | + 0.500·5-s + 0.416·7-s + 0.368·11-s + 0.397·13-s + 0.177·17-s − 0.0394·19-s + 0.909·23-s − 0.832·25-s + 0.529·29-s − 0.180·31-s + 0.208·35-s − 0.982·37-s + 1.78·41-s + 0.929·43-s + 1.85·47-s − 1.14·49-s + 2.29·53-s + 0.184·55-s + 0.777·59-s − 0.404·61-s + 0.198·65-s + 0.939·67-s + 2.42·71-s − 1.23·73-s + 0.153·77-s + 0.288·79-s + 1.37·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.090858194\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.090858194\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 28 T + 3386 T^{2} - 28 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 54 T + 22103 T^{2} - 54 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 148 T + 300038 T^{2} - 148 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 242 T + 451227 T^{2} - 242 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 212 T + 2333810 T^{2} - 212 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 62 T + 4757319 T^{2} + 62 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2308 T + 12585662 T^{2} - 2308 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2400 T + 18765658 T^{2} - 2400 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 968 T + 33795918 T^{2} + 968 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8178 T + 40241675 T^{2} + 8178 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 19200 T + 245536402 T^{2} - 19200 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 11264 T + 155306550 T^{2} - 11264 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 28116 T + 642581038 T^{2} - 28116 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 46984 T + 1367689610 T^{2} - 46984 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 20788 T + 264847334 T^{2} - 20788 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 11770 T + 1507312467 T^{2} + 11770 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 34522 T + 2997408975 T^{2} - 34522 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 103016 T + 6203258126 T^{2} - 103016 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 56158 T + 4934132787 T^{2} + 56158 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 16010 T + 870279783 T^{2} - 16010 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 86216 T + 7112504390 T^{2} - 86216 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 172164 T + 17420861122 T^{2} - 172164 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 139430 T + 17569513899 T^{2} + 139430 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−11.66303454691228787326764181209, −11.15620112863046134423460516300, −10.70275190541978142215598243017, −10.35206447901050063877918803028, −9.627791469220645456976365288927, −9.277804031621697438411646135564, −8.747195036506084088083514396980, −8.332785784284408383653299802884, −7.47988964371922424709763408438, −7.35303518869632807555835312966, −6.39179155688824271620678734732, −6.15462206835493681802402422405, −5.29402484616183987080426182399, −5.07138241447081236348468950869, −3.95085446561899152058034703208, −3.81986713002362946301806720411, −2.63854789440162985223314086523, −2.18376302998332980039128728629, −1.21520818831689434079128578298, −0.67899313635227505794628865887,
0.67899313635227505794628865887, 1.21520818831689434079128578298, 2.18376302998332980039128728629, 2.63854789440162985223314086523, 3.81986713002362946301806720411, 3.95085446561899152058034703208, 5.07138241447081236348468950869, 5.29402484616183987080426182399, 6.15462206835493681802402422405, 6.39179155688824271620678734732, 7.35303518869632807555835312966, 7.47988964371922424709763408438, 8.332785784284408383653299802884, 8.747195036506084088083514396980, 9.277804031621697438411646135564, 9.627791469220645456976365288927, 10.35206447901050063877918803028, 10.70275190541978142215598243017, 11.15620112863046134423460516300, 11.66303454691228787326764181209
