L(s) = 1 | + 2-s + 3·3-s + 3·5-s + 3·6-s + 2·8-s + 6·9-s + 3·10-s + 2·13-s + 9·15-s + 3·16-s + 2·17-s + 6·18-s − 8·19-s + 6·24-s + 6·25-s + 2·26-s + 10·27-s + 10·29-s + 9·30-s + 8·31-s + 3·32-s + 2·34-s − 6·37-s − 8·38-s + 6·39-s + 6·40-s + 14·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.73·3-s + 1.34·5-s + 1.22·6-s + 0.707·8-s + 2·9-s + 0.948·10-s + 0.554·13-s + 2.32·15-s + 3/4·16-s + 0.485·17-s + 1.41·18-s − 1.83·19-s + 1.22·24-s + 6/5·25-s + 0.392·26-s + 1.92·27-s + 1.85·29-s + 1.64·30-s + 1.43·31-s + 0.530·32-s + 0.342·34-s − 0.986·37-s − 1.29·38-s + 0.960·39-s + 0.948·40-s + 2.18·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(18.03420967\) |
\(L(\frac12)\) |
\(\approx\) |
\(18.03420967\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| 11 | | \( 1 \) |
good | 2 | $S_4\times C_2$ | \( 1 - T + T^{2} - 3 T^{3} + p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 5 T^{2} - 16 T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 13 | $D_{6}$ | \( 1 - 2 T + 27 T^{2} - 44 T^{3} + 27 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 2 T - T^{2} + 116 T^{3} - p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 8 T + 41 T^{2} + 144 T^{3} + 41 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 5 T^{2} - 128 T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 10 T + 99 T^{2} - 540 T^{3} + 99 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 8 T + 61 T^{2} - 368 T^{3} + 61 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 41 | $S_4\times C_2$ | \( 1 - 14 T + 167 T^{2} - 1156 T^{3} + 167 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 4 T + 49 T^{2} - 56 T^{3} + 49 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 8 T + 109 T^{2} + 624 T^{3} + 109 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 6 T + 107 T^{2} + 644 T^{3} + 107 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 12 T + 161 T^{2} - 1096 T^{3} + 161 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 6 T + 131 T^{2} - 484 T^{3} + 131 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 4 T + 153 T^{2} + 472 T^{3} + 153 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 8 T + 181 T^{2} - 1008 T^{3} + 181 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 14 T + 223 T^{2} - 1700 T^{3} + 223 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 12 T + 173 T^{2} + 1096 T^{3} + 173 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 129 T^{2} - 16 T^{3} + 129 p T^{4} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 10 T + 215 T^{2} + 1580 T^{3} + 215 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 22 T + 399 T^{2} - 4276 T^{3} + 399 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.417074726670171228732884791717, −8.064980757721626144435657038888, −7.87032407848544047831295562515, −7.30969757836230906365548789429, −7.27459501172959312068476464470, −6.88161925653178687353406807453, −6.40870015602004296514712804773, −6.37803453785360511975715515429, −6.18784451316645286782615283200, −5.96303005005814666030707535135, −5.21771802521927821732912709099, −5.04916161274286318029216841143, −5.01621108430893583738914439387, −4.49712345373581957621470083778, −4.17949852822205665608011889483, −4.14849204192772368969300782728, −3.61665316948219825277698527930, −3.33565697388660835294905882749, −2.95190567342759080754221366648, −2.54783851640045276144847827202, −2.49282677479250398565505451754, −1.84048687588972411031430750988, −1.81019927113047872090365336656, −1.14030064010705934337546709362, −0.832755459395975989292325862828,
0.832755459395975989292325862828, 1.14030064010705934337546709362, 1.81019927113047872090365336656, 1.84048687588972411031430750988, 2.49282677479250398565505451754, 2.54783851640045276144847827202, 2.95190567342759080754221366648, 3.33565697388660835294905882749, 3.61665316948219825277698527930, 4.14849204192772368969300782728, 4.17949852822205665608011889483, 4.49712345373581957621470083778, 5.01621108430893583738914439387, 5.04916161274286318029216841143, 5.21771802521927821732912709099, 5.96303005005814666030707535135, 6.18784451316645286782615283200, 6.37803453785360511975715515429, 6.40870015602004296514712804773, 6.88161925653178687353406807453, 7.27459501172959312068476464470, 7.30969757836230906365548789429, 7.87032407848544047831295562515, 8.064980757721626144435657038888, 8.417074726670171228732884791717