| L(s) = 1 | + 8·2-s − 6·3-s − 148·4-s − 470·5-s − 48·6-s + 1.22e3·7-s − 1.85e3·8-s − 2.18e3·9-s − 3.76e3·10-s + 888·12-s − 344·13-s + 9.82e3·14-s + 2.82e3·15-s + 8.33e3·16-s + 8.46e3·17-s − 1.74e4·18-s + 3.52e4·19-s + 6.95e4·20-s − 7.36e3·21-s − 6.14e4·23-s + 1.11e4·24-s + 3.34e4·25-s − 2.75e3·26-s + 1.33e4·27-s − 1.81e5·28-s − 1.79e5·29-s + 2.25e4·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.128·3-s − 1.15·4-s − 1.68·5-s − 0.0907·6-s + 1.35·7-s − 1.28·8-s − 9-s − 1.18·10-s + 0.148·12-s − 0.0434·13-s + 0.956·14-s + 0.215·15-s + 0.508·16-s + 0.418·17-s − 0.707·18-s + 1.18·19-s + 1.94·20-s − 0.173·21-s − 1.05·23-s + 0.164·24-s + 0.427·25-s − 0.0307·26-s + 0.130·27-s − 1.56·28-s − 1.36·29-s + 0.152·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14641 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14641 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 11 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - p^{3} T + 53 p^{2} T^{2} - p^{10} T^{3} + p^{14} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + 2 p T + 247 p^{2} T^{2} + 2 p^{8} T^{3} + p^{14} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 94 p T + 7499 p^{2} T^{2} + 94 p^{8} T^{3} + p^{14} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 1228 T + 1620642 T^{2} - 1228 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 344 T + 109427178 T^{2} + 344 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 8468 T + 1895926 p T^{2} - 8468 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 35280 T + 1565373638 T^{2} - 35280 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 61486 T + 6650536943 T^{2} + 61486 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 179040 T + 34622681578 T^{2} + 179040 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 57166 T + 31335570111 T^{2} + 57166 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 877698 T + 381609348827 T^{2} + 877698 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 283616 T + 125033442626 T^{2} - 283616 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 275484 T + 552841024778 T^{2} + 275484 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 1662512 T + 1690275368222 T^{2} - 1662512 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 1616484 T + 2738235417598 T^{2} - 1616484 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 2454130 T + 5185075198823 T^{2} + 2454130 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6019176 T + 15292993001786 T^{2} - 6019176 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 174698 T + 2881438572447 T^{2} + 174698 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1151466 T + 6215147678071 T^{2} + 1151466 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 885944 T + 20004042345618 T^{2} + 885944 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 3801460 T + 35726075376258 T^{2} + 3801460 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2282916 T + 53736100067578 T^{2} - 2282916 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 13481970 T + 131461606905283 T^{2} + 13481970 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 68078 T + 159761571363987 T^{2} + 68078 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−11.76476085563986157917656659298, −11.62942812314855181446146891100, −11.08542039668236334959318056603, −10.37790094356542353681771905902, −9.673987524020814628768759609390, −8.975930619489815812955867369036, −8.360383450996119447022591710883, −8.322739109786200603382980220602, −7.39260057071831096951729433467, −7.28716449374595660587140684928, −5.67559493507687863935326131421, −5.57821410904427118192153591301, −4.95666564803320549111228116189, −4.24765520760513660120644065518, −3.71456026595433331926180411456, −3.44684350157786579715326917153, −2.18358884558059466152211341764, −1.09448653266643213527323513432, 0, 0,
1.09448653266643213527323513432, 2.18358884558059466152211341764, 3.44684350157786579715326917153, 3.71456026595433331926180411456, 4.24765520760513660120644065518, 4.95666564803320549111228116189, 5.57821410904427118192153591301, 5.67559493507687863935326131421, 7.28716449374595660587140684928, 7.39260057071831096951729433467, 8.322739109786200603382980220602, 8.360383450996119447022591710883, 8.975930619489815812955867369036, 9.673987524020814628768759609390, 10.37790094356542353681771905902, 11.08542039668236334959318056603, 11.62942812314855181446146891100, 11.76476085563986157917656659298
