| L(s) = 1 | − 3·2-s + 3·3-s + 6·4-s − 9·6-s − 10·8-s + 6·9-s + 8·11-s + 18·12-s − 4·13-s + 15·16-s − 2·17-s − 18·18-s + 3·19-s − 24·22-s + 8·23-s − 30·24-s + 12·26-s + 10·27-s + 10·29-s − 21·32-s + 24·33-s + 6·34-s + 36·36-s − 4·37-s − 9·38-s − 12·39-s + 4·41-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 1.73·3-s + 3·4-s − 3.67·6-s − 3.53·8-s + 2·9-s + 2.41·11-s + 5.19·12-s − 1.10·13-s + 15/4·16-s − 0.485·17-s − 4.24·18-s + 0.688·19-s − 5.11·22-s + 1.66·23-s − 6.12·24-s + 2.35·26-s + 1.92·27-s + 1.85·29-s − 3.71·32-s + 4.17·33-s + 1.02·34-s + 6·36-s − 0.657·37-s − 1.45·38-s − 1.92·39-s + 0.624·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.737974724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.737974724\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + T )^{3} \) |
| 3 | $C_1$ | \( ( 1 - T )^{3} \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 + 5 T^{2} - 16 T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 8 T + 41 T^{2} - 160 T^{3} + 41 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 4 T + 23 T^{2} + 72 T^{3} + 23 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 2 T + 31 T^{2} + 60 T^{3} + 31 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 8 T + 77 T^{2} - 352 T^{3} + 77 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 10 T + 107 T^{2} - 572 T^{3} + 107 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 77 T^{2} + 16 T^{3} + 77 p T^{4} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 4 T + 95 T^{2} + 264 T^{3} + 95 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 4 T + 43 T^{2} + 72 T^{3} + 43 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 6 T + 5 T^{2} + 244 T^{3} + 5 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 8 T + 149 T^{2} + 736 T^{3} + 149 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{3} \) |
| 59 | $S_4\times C_2$ | \( 1 - 10 T + 173 T^{2} - 1172 T^{3} + 173 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 14 T + 195 T^{2} - 1556 T^{3} + 195 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 20 T + 249 T^{2} + 2360 T^{3} + 249 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 20 T + 261 T^{2} - 2520 T^{3} + 261 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 8 T + 155 T^{2} + 912 T^{3} + 155 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 221 T^{2} - 16 T^{3} + 221 p T^{4} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 4 T + 73 T^{2} - 504 T^{3} + 73 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 24 T + 443 T^{2} - 4672 T^{3} + 443 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 22 T + 439 T^{2} + 4564 T^{3} + 439 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.935469111909716356592842401939, −7.54063213240306111497977594271, −7.25975710478639880715684633191, −7.22359857185165900304324180377, −6.91996539698461986412585181199, −6.75549680430511390521248925128, −6.61062425107664451883093656253, −6.15441964035426648046034271803, −5.95377223316173105474604772884, −5.62295533242815469924388849220, −4.89984202987099233873262723358, −4.89333896952118157184148507713, −4.73080072424621857079550472221, −4.04272988404472190468152791959, −3.81615158112852592072856875039, −3.74346983946274396758002908504, −3.10806321221570367114970203116, −2.95165243660714272399932394471, −2.82725887838524761361023455161, −2.19640903205882778622603303355, −1.94965382314228903821866347966, −1.89636841695441668962170976633, −1.05209755397635372022173710688, −0.969173666826125565904573100662, −0.70465331463598569737567360346,
0.70465331463598569737567360346, 0.969173666826125565904573100662, 1.05209755397635372022173710688, 1.89636841695441668962170976633, 1.94965382314228903821866347966, 2.19640903205882778622603303355, 2.82725887838524761361023455161, 2.95165243660714272399932394471, 3.10806321221570367114970203116, 3.74346983946274396758002908504, 3.81615158112852592072856875039, 4.04272988404472190468152791959, 4.73080072424621857079550472221, 4.89333896952118157184148507713, 4.89984202987099233873262723358, 5.62295533242815469924388849220, 5.95377223316173105474604772884, 6.15441964035426648046034271803, 6.61062425107664451883093656253, 6.75549680430511390521248925128, 6.91996539698461986412585181199, 7.22359857185165900304324180377, 7.25975710478639880715684633191, 7.54063213240306111497977594271, 7.935469111909716356592842401939
