| L(s) = 1 | + 3·2-s + 3·3-s + 6·4-s + 9·6-s + 10·8-s + 6·9-s + 3·11-s + 18·12-s + 3·13-s + 15·16-s + 6·17-s + 18·18-s + 3·19-s + 9·22-s + 9·23-s + 30·24-s + 9·26-s + 10·27-s + 12·29-s + 21·32-s + 9·33-s + 18·34-s + 36·36-s + 9·37-s + 9·38-s + 9·39-s − 9·41-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 + 0.904·11-s + 5.19·12-s + 0.832·13-s + 15/4·16-s + 1.45·17-s + 4.24·18-s + 0.688·19-s + 1.91·22-s + 1.87·23-s + 6.12·24-s + 1.76·26-s + 1.92·27-s + 2.22·29-s + 3.71·32-s + 1.56·33-s + 3.08·34-s + 6·36-s + 1.47·37-s + 1.45·38-s + 1.44·39-s − 1.40·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(98.11345967\) |
| \(L(\frac12)\) |
\(\approx\) |
\(98.11345967\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 11 | $S_4\times C_2$ | \( 1 - 3 T + 6 T^{2} - 17 T^{3} + 6 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 3 T + 27 T^{2} - 54 T^{3} + 27 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 6 T + 3 T^{2} + 4 p T^{3} + 3 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 3 T + 45 T^{2} - 90 T^{3} + 45 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 9 T + 81 T^{2} - 406 T^{3} + 81 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 12 T + 120 T^{2} - 720 T^{3} + 120 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 78 T^{2} - 10 T^{3} + 78 p T^{4} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 9 T + 123 T^{2} - 658 T^{3} + 123 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 9 T + 75 T^{2} + 610 T^{3} + 75 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 47 | $S_4\times C_2$ | \( 1 + 3 T + 9 T^{2} - 122 T^{3} + 9 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 9 T + 66 T^{2} + 141 T^{3} + 66 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 12 T + 150 T^{2} + 980 T^{3} + 150 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 6 T + 75 T^{2} - 20 T^{3} + 75 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 6 T + 93 T^{2} + 412 T^{3} + 93 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{3} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{3} \) |
| 79 | $S_4\times C_2$ | \( 1 - 24 T + 414 T^{2} - 4194 T^{3} + 414 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 12 T + 222 T^{2} - 1906 T^{3} + 222 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 6 T + 159 T^{2} + 676 T^{3} + 159 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 24 T + 408 T^{2} + 4768 T^{3} + 408 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−6.85786886197089372144910122844, −6.63785680039348900077845524215, −6.54275822909992105662222722925, −6.54047211443070837462791610159, −5.96009295217820891406160414244, −5.64282010010762836968719655500, −5.62300045209755651968906724899, −5.20152997788629809615245533459, −5.00143518303588654245870807887, −4.77536587018160235848992072443, −4.46540755473213992444800790998, −4.33747919264829024134419707397, −4.14470733408602313227447182567, −3.60831987896647723848527591607, −3.38866095664908052484563711517, −3.38644200586490111580477367807, −3.15910025287433846366997008447, −2.86753468223265788787549117081, −2.78753895184083784276427500983, −2.19477984177765303675755027586, −1.91200915857080450188965801961, −1.77918240506353131733186977027, −1.11950575736445654666377926288, −1.05368137191529489270910606001, −0.854488245661162018636994088215,
0.854488245661162018636994088215, 1.05368137191529489270910606001, 1.11950575736445654666377926288, 1.77918240506353131733186977027, 1.91200915857080450188965801961, 2.19477984177765303675755027586, 2.78753895184083784276427500983, 2.86753468223265788787549117081, 3.15910025287433846366997008447, 3.38644200586490111580477367807, 3.38866095664908052484563711517, 3.60831987896647723848527591607, 4.14470733408602313227447182567, 4.33747919264829024134419707397, 4.46540755473213992444800790998, 4.77536587018160235848992072443, 5.00143518303588654245870807887, 5.20152997788629809615245533459, 5.62300045209755651968906724899, 5.64282010010762836968719655500, 5.96009295217820891406160414244, 6.54047211443070837462791610159, 6.54275822909992105662222722925, 6.63785680039348900077845524215, 6.85786886197089372144910122844
