| L(s) = 1 | + 9·2-s + 6·3-s + 11·4-s + 18·5-s + 54·6-s − 243·8-s + 54·9-s + 162·10-s + 396·11-s + 66·12-s + 350·13-s + 108·15-s − 1.38e3·16-s − 1.80e3·17-s + 486·18-s + 3.26e3·19-s + 198·20-s + 3.56e3·22-s + 2.08e3·23-s − 1.45e3·24-s − 4.58e3·25-s + 3.15e3·26-s + 1.89e3·27-s + 6.69e3·29-s + 972·30-s + 20·31-s − 1.53e3·32-s + ⋯ |
| L(s) = 1 | + 1.59·2-s + 0.384·3-s + 0.343·4-s + 0.321·5-s + 0.612·6-s − 1.34·8-s + 2/9·9-s + 0.512·10-s + 0.986·11-s + 0.132·12-s + 0.574·13-s + 0.123·15-s − 1.35·16-s − 1.51·17-s + 0.353·18-s + 2.07·19-s + 0.110·20-s + 1.56·22-s + 0.823·23-s − 0.516·24-s − 1.46·25-s + 0.913·26-s + 0.498·27-s + 1.47·29-s + 0.197·30-s + 0.00373·31-s − 0.265·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.332497585\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.332497585\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - 9 T + 35 p T^{2} - 9 p^{5} T^{3} + p^{10} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - 2 p T - 2 p^{2} T^{2} - 2 p^{6} T^{3} + p^{10} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 18 T + 4906 T^{2} - 18 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 36 p T + 142198 T^{2} - 36 p^{6} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 350 T + 546978 T^{2} - 350 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 1800 T + 3567406 T^{2} + 1800 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 3266 T + 7614270 T^{2} - 3266 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2088 T + 9365230 T^{2} - 2088 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6696 T + 51326470 T^{2} - 6696 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 20 T + 53103102 T^{2} - 20 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6232 T + 144242070 T^{2} - 6232 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6048 T + 223864366 T^{2} - 6048 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3020 T - 30383466 T^{2} + 3020 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 11700 T + 292735582 T^{2} + 11700 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 9468 T + 858185230 T^{2} - 9468 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 43938 T + 1852599934 T^{2} - 43938 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 64754 T + 2408321418 T^{2} - 64754 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 24784 T + 2799959190 T^{2} - 24784 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 97416 T + 5729557966 T^{2} - 97416 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 17452 T + 3828622374 T^{2} + 17452 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 51256 T + 3645565854 T^{2} - 51256 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 117558 T + 7798161502 T^{2} + 117558 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 84276 T + 5915697430 T^{2} + 84276 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 20776 T + 16174049358 T^{2} + 20776 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−14.50650931889253454991758887742, −14.25098854633383623965881929816, −13.53836852114974685457043429737, −13.53294464633211812073556188810, −12.85004302132542152400669724060, −12.21327153342058952800485451861, −11.50487530191754428491634566447, −11.19735090778495609415636323882, −9.802350256584373282173163782327, −9.651982442170722786160835903192, −8.779011300765962938663648196555, −8.289595929222964735770111160818, −7.06386321099988479912435345948, −6.47711560616122517865296294568, −5.60520065657440071078627447761, −4.97034185271461722802199054572, −4.12898036252271703377730697862, −3.60370576816232366154835316369, −2.56382232392017197037896481000, −1.00968748692667881084536373013,
1.00968748692667881084536373013, 2.56382232392017197037896481000, 3.60370576816232366154835316369, 4.12898036252271703377730697862, 4.97034185271461722802199054572, 5.60520065657440071078627447761, 6.47711560616122517865296294568, 7.06386321099988479912435345948, 8.289595929222964735770111160818, 8.779011300765962938663648196555, 9.651982442170722786160835903192, 9.802350256584373282173163782327, 11.19735090778495609415636323882, 11.50487530191754428491634566447, 12.21327153342058952800485451861, 12.85004302132542152400669724060, 13.53294464633211812073556188810, 13.53836852114974685457043429737, 14.25098854633383623965881929816, 14.50650931889253454991758887742
