| 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
| \(L(1)\) |
\(\approx\) |
\(0.5664071394\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5664071394\) |
| \(L(\frac{3}{2})\) |
|
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
−7.11788812149158815281449905330, −6.71297822941553471835126847894, −6.67503177093815258263787547267, −6.44318277124310238104848376374, −6.06336663583584330009752630683, −5.95409394286518012476969506343, −5.89728084357271610033423030802, −5.29832100085050992452041840826, −5.14180602346829976497170300138, −5.06944160856705618755249813651, −4.51128520063748240406843190568, −4.44684049976168708762374244739, −4.24551523368724188291629497263, −3.60061949402398847487851779337, −3.49711547500321152888942927592, −3.48490080427250521537798959424, −2.57315359897174861384693834646, −2.46827835199970228117572843532, −2.43781346352175813417460558488, −1.74909383590853446842109233220, −1.68817005675642788485807164296, −1.38872773927575195555303817366, −0.821140754681115451691950261728, −0.48561816343868350318216596720, −0.39771563005368624486744879396,
0.39771563005368624486744879396, 0.48561816343868350318216596720, 0.821140754681115451691950261728, 1.38872773927575195555303817366, 1.68817005675642788485807164296, 1.74909383590853446842109233220, 2.43781346352175813417460558488, 2.46827835199970228117572843532, 2.57315359897174861384693834646, 3.48490080427250521537798959424, 3.49711547500321152888942927592, 3.60061949402398847487851779337, 4.24551523368724188291629497263, 4.44684049976168708762374244739, 4.51128520063748240406843190568, 5.06944160856705618755249813651, 5.14180602346829976497170300138, 5.29832100085050992452041840826, 5.89728084357271610033423030802, 5.95409394286518012476969506343, 6.06336663583584330009752630683, 6.44318277124310238104848376374, 6.67503177093815258263787547267, 6.71297822941553471835126847894, 7.11788812149158815281449905330
