| L(s) = 1 | − 2·2-s − 67·4-s − 75·5-s − 68·7-s + 166·8-s + 150·10-s − 363·11-s + 290·13-s + 136·14-s + 2.37e3·16-s − 434·17-s − 2.85e3·19-s + 5.02e3·20-s + 726·22-s + 640·23-s + 3.75e3·25-s − 580·26-s + 4.55e3·28-s + 4.53e3·29-s − 1.49e4·31-s − 6.66e3·32-s + 868·34-s + 5.10e3·35-s − 6.19e3·37-s + 5.71e3·38-s − 1.24e4·40-s + 8.92e3·41-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 2.09·4-s − 1.34·5-s − 0.524·7-s + 0.917·8-s + 0.474·10-s − 0.904·11-s + 0.475·13-s + 0.185·14-s + 2.31·16-s − 0.364·17-s − 1.81·19-s + 2.80·20-s + 0.319·22-s + 0.252·23-s + 6/5·25-s − 0.168·26-s + 1.09·28-s + 1.00·29-s − 2.79·31-s − 1.15·32-s + 0.128·34-s + 0.703·35-s − 0.743·37-s + 0.641·38-s − 1.23·40-s + 0.829·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.7338339493\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7338339493\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p^{2} T )^{3} \) |
| 11 | $C_1$ | \( ( 1 + p^{2} T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + p T + 71 T^{2} + 55 p T^{3} + 71 p^{5} T^{4} + p^{11} T^{5} + p^{15} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 68 T + 34869 T^{2} + 2533560 T^{3} + 34869 p^{5} T^{4} + 68 p^{10} T^{5} + p^{15} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 290 T + 658819 T^{2} - 83286348 T^{3} + 658819 p^{5} T^{4} - 290 p^{10} T^{5} + p^{15} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 434 T + 58991 T^{2} - 1314616612 T^{3} + 58991 p^{5} T^{4} + 434 p^{10} T^{5} + p^{15} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 2856 T + 8753113 T^{2} + 14281181168 T^{3} + 8753113 p^{5} T^{4} + 2856 p^{10} T^{5} + p^{15} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 640 T + 6700261 T^{2} + 7538830592 T^{3} + 6700261 p^{5} T^{4} - 640 p^{10} T^{5} + p^{15} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 4538 T + 35092643 T^{2} - 141745639868 T^{3} + 35092643 p^{5} T^{4} - 4538 p^{10} T^{5} + p^{15} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 14968 T + 160071261 T^{2} + 978687787280 T^{3} + 160071261 p^{5} T^{4} + 14968 p^{10} T^{5} + p^{15} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 6190 T + 113089771 T^{2} + 886491849396 T^{3} + 113089771 p^{5} T^{4} + 6190 p^{10} T^{5} + p^{15} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 8926 T + 272145047 T^{2} - 2187570068644 T^{3} + 272145047 p^{5} T^{4} - 8926 p^{10} T^{5} + p^{15} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 33592 T + 693464257 T^{2} + 9837617686992 T^{3} + 693464257 p^{5} T^{4} + 33592 p^{10} T^{5} + p^{15} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 24640 T + 788101261 T^{2} - 10622124840832 T^{3} + 788101261 p^{5} T^{4} - 24640 p^{10} T^{5} + p^{15} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 22934 T + 1099334651 T^{2} - 14788031800868 T^{3} + 1099334651 p^{5} T^{4} - 22934 p^{10} T^{5} + p^{15} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 13756 T - 129165359 T^{2} + 1129007325848 T^{3} - 129165359 p^{5} T^{4} - 13756 p^{10} T^{5} + p^{15} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 24602 T + 2638007155 T^{2} - 41514451599900 T^{3} + 2638007155 p^{5} T^{4} - 24602 p^{10} T^{5} + p^{15} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 16868 T + 3843672649 T^{2} - 43721067889560 T^{3} + 3843672649 p^{5} T^{4} - 16868 p^{10} T^{5} + p^{15} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 4856 T + 1951490165 T^{2} - 16632390591088 T^{3} + 1951490165 p^{5} T^{4} + 4856 p^{10} T^{5} + p^{15} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 1910 T + 5875530055 T^{2} - 8082237219348 T^{3} + 5875530055 p^{5} T^{4} - 1910 p^{10} T^{5} + p^{15} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 36844 T + 4302241965 T^{2} + 155969878390184 T^{3} + 4302241965 p^{5} T^{4} + 36844 p^{10} T^{5} + p^{15} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 48796 T + 9666032953 T^{2} - 314032777245928 T^{3} + 9666032953 p^{5} T^{4} - 48796 p^{10} T^{5} + p^{15} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 188978 T + 25306287767 T^{2} - 2094440123430812 T^{3} + 25306287767 p^{5} T^{4} - 188978 p^{10} T^{5} + p^{15} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 247526 T + 41254986031 T^{2} - 4400049090000852 T^{3} + 41254986031 p^{5} T^{4} - 247526 p^{10} T^{5} + p^{15} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.086253849255547889045641472559, −8.469015215439844258900122145900, −8.464889724283013750176615172997, −8.383053286391489149839130517579, −7.77979711033930729970942349196, −7.60073584486148454846809951154, −7.26348860449073221158937344518, −6.75875940176293052714813464676, −6.59704470162356467193931530052, −6.19709274246448217644256634064, −5.52667564638945336117210039116, −5.48942556043060736269082089962, −4.91309294952406055387739865405, −4.78624471986705496006413912244, −4.36400482999512249024689398846, −4.12834240531880298761248803484, −3.65106973807353494712077878195, −3.41375113520436924607620786305, −3.26354768190471446030513561848, −2.44750071205673160746341924653, −1.98130498614868975564402576676, −1.60921500684387199888985101616, −0.62377404497088851152038135075, −0.45238787643996408916638723705, −0.41210886374434186777067571915,
0.41210886374434186777067571915, 0.45238787643996408916638723705, 0.62377404497088851152038135075, 1.60921500684387199888985101616, 1.98130498614868975564402576676, 2.44750071205673160746341924653, 3.26354768190471446030513561848, 3.41375113520436924607620786305, 3.65106973807353494712077878195, 4.12834240531880298761248803484, 4.36400482999512249024689398846, 4.78624471986705496006413912244, 4.91309294952406055387739865405, 5.48942556043060736269082089962, 5.52667564638945336117210039116, 6.19709274246448217644256634064, 6.59704470162356467193931530052, 6.75875940176293052714813464676, 7.26348860449073221158937344518, 7.60073584486148454846809951154, 7.77979711033930729970942349196, 8.383053286391489149839130517579, 8.464889724283013750176615172997, 8.469015215439844258900122145900, 9.086253849255547889045641472559
