| L(s) = 1 | + 3·3-s − 6·4-s + 6·9-s − 18·12-s + 3·13-s + 24·16-s + 6·17-s + 9·19-s + 6·23-s − 3·25-s + 10·27-s − 3·31-s − 36·36-s − 3·37-s + 9·39-s + 3·41-s − 21·43-s + 72·48-s + 18·51-s − 18·52-s − 6·53-s + 27·57-s − 18·59-s − 6·61-s − 80·64-s + 3·67-s − 36·68-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 3·4-s + 2·9-s − 5.19·12-s + 0.832·13-s + 6·16-s + 1.45·17-s + 2.06·19-s + 1.25·23-s − 3/5·25-s + 1.92·27-s − 0.538·31-s − 6·36-s − 0.493·37-s + 1.44·39-s + 0.468·41-s − 3.20·43-s + 10.3·48-s + 2.52·51-s − 2.49·52-s − 0.824·53-s + 3.57·57-s − 2.34·59-s − 0.768·61-s − 10·64-s + 0.366·67-s − 4.36·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{6} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{6} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.918876737\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.918876737\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 7 | | \( 1 \) |
| 41 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 5 | $S_4\times C_2$ | \( 1 + 3 T^{2} - 14 T^{3} + 3 p T^{4} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 3 T^{2} - 50 T^{3} + 3 p T^{4} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 3 T + 24 T^{2} - 43 T^{3} + 24 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 19 | $S_4\times C_2$ | \( 1 - 9 T + 72 T^{2} - 347 T^{3} + 72 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 6 T + 3 p T^{2} - 274 T^{3} + 3 p^{2} T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 39 T^{2} - 112 T^{3} + 39 p T^{4} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 3 T + 24 T^{2} + 79 T^{3} + 24 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{3} \) |
| 43 | $S_4\times C_2$ | \( 1 + 21 T + 264 T^{2} + 2069 T^{3} + 264 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 39 T^{2} + 274 T^{3} + 39 p T^{4} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 6 T + 99 T^{2} + 356 T^{3} + 99 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 18 T + 213 T^{2} + 1764 T^{3} + 213 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 6 T + 75 T^{2} + 75 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 3 T + 186 T^{2} - 403 T^{3} + 186 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 141 T^{2} + 144 T^{3} + 141 p T^{4} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 21 T + 354 T^{2} - 3329 T^{3} + 354 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 21 T + 312 T^{2} - 3391 T^{3} + 312 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 6 T + 81 T^{2} - 194 T^{3} + 81 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 12 T + 213 T^{2} + 1518 T^{3} + 213 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 18 T + 207 T^{2} + 1660 T^{3} + 207 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.54422487480498289736370220263, −6.94990215325384962112358369114, −6.71694844932871923206438935852, −6.55919137644706151574985763804, −6.02984364808429978834508874751, −5.71542062277726249093960418151, −5.68627742888609876906107827998, −5.26996102695663026330257622741, −5.12309004372907856619267532386, −4.93435940596202892189719674817, −4.68501963640930758380580806774, −4.38552417334536899619247868048, −4.20205410891521427056365964856, −3.72032357626640327382059605224, −3.64759605908625853026487173457, −3.27736178054778920499471175059, −3.22286313169863185415715864309, −3.02603054300320608141869903798, −2.98729320808544477746515686496, −1.88635662204802283836532648170, −1.73961647573708529841781088043, −1.63559194624772341383681148008, −0.996681064723508910043302506667, −0.72624118072804194287913613433, −0.52514240811785436406019801809,
0.52514240811785436406019801809, 0.72624118072804194287913613433, 0.996681064723508910043302506667, 1.63559194624772341383681148008, 1.73961647573708529841781088043, 1.88635662204802283836532648170, 2.98729320808544477746515686496, 3.02603054300320608141869903798, 3.22286313169863185415715864309, 3.27736178054778920499471175059, 3.64759605908625853026487173457, 3.72032357626640327382059605224, 4.20205410891521427056365964856, 4.38552417334536899619247868048, 4.68501963640930758380580806774, 4.93435940596202892189719674817, 5.12309004372907856619267532386, 5.26996102695663026330257622741, 5.68627742888609876906107827998, 5.71542062277726249093960418151, 6.02984364808429978834508874751, 6.55919137644706151574985763804, 6.71694844932871923206438935852, 6.94990215325384962112358369114, 7.54422487480498289736370220263
