| L(s) = 1 | + 3-s + 3·7-s − 9-s + 3·11-s + 3·13-s − 5·17-s + 7·19-s + 3·21-s + 10·23-s − 3·27-s + 10·29-s − 3·31-s + 3·33-s − 8·37-s + 3·39-s + 4·41-s + 9·43-s − 49-s − 5·51-s + 5·53-s + 7·57-s − 61-s − 3·63-s − 14·67-s + 10·69-s + 17·71-s − 21·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.13·7-s − 1/3·9-s + 0.904·11-s + 0.832·13-s − 1.21·17-s + 1.60·19-s + 0.654·21-s + 2.08·23-s − 0.577·27-s + 1.85·29-s − 0.538·31-s + 0.522·33-s − 1.31·37-s + 0.480·39-s + 0.624·41-s + 1.37·43-s − 1/7·49-s − 0.700·51-s + 0.686·53-s + 0.927·57-s − 0.128·61-s − 0.377·63-s − 1.71·67-s + 1.20·69-s + 2.01·71-s − 2.45·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.590135947\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.590135947\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - T + 2 T^{2} + 2 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 3 T + 10 T^{2} - 10 T^{3} + 10 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{3} \) |
| 17 | $S_4\times C_2$ | \( 1 + 5 T + 52 T^{2} + 166 T^{3} + 52 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 7 T + 44 T^{2} - 239 T^{3} + 44 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 10 T + 95 T^{2} - 469 T^{3} + 95 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 10 T + 79 T^{2} - 379 T^{3} + 79 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 3 T + 40 T^{2} - 21 T^{3} + 40 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 8 T + 110 T^{2} + 548 T^{3} + 110 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 4 T + 106 T^{2} - 304 T^{3} + 106 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 9 T + 73 T^{2} - 266 T^{3} + 73 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 64 T^{2} + 192 T^{3} + 64 p T^{4} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 5 T + 18 T^{2} - 388 T^{3} + 18 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 100 T^{2} + 192 T^{3} + 100 p T^{4} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + T + 142 T^{2} + 8 T^{3} + 142 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 14 T + 185 T^{2} + 1780 T^{3} + 185 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 17 T + 287 T^{2} - 2466 T^{3} + 287 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 21 T + 240 T^{2} + 2014 T^{3} + 240 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 11 T + 216 T^{2} - 1594 T^{3} + 216 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 20 T + 255 T^{2} - 2491 T^{3} + 255 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 30 T + 553 T^{2} - 6219 T^{3} + 553 p T^{4} - 30 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 10 T + 259 T^{2} + 1917 T^{3} + 259 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.282238517848677303279927684650, −7.60486624881837416323507595250, −7.55683862445523839866638530289, −7.26209424892157839975387495577, −7.19068697584096815569796255976, −6.69660541094886013523918124332, −6.50279266457793012599982116282, −6.19744656518248126120863374994, −5.85771379787247665333969967904, −5.78849425932937401148045538510, −5.07248697627071689365880182928, −5.06531494702112833482969131234, −4.84037565255270880140258086995, −4.59525776976627305994409436949, −4.04400906574502903625827291092, −4.00086460011873360278131875089, −3.38348288576375195770306636875, −3.20995089512368863973183995384, −3.09483146392753127731384708748, −2.52672092176313107929851489870, −2.16793981535468247440050763042, −1.83236932153670231053291674231, −1.40771600723921127728975150122, −0.816851140987039981840094406086, −0.801249045547959097899904315865,
0.801249045547959097899904315865, 0.816851140987039981840094406086, 1.40771600723921127728975150122, 1.83236932153670231053291674231, 2.16793981535468247440050763042, 2.52672092176313107929851489870, 3.09483146392753127731384708748, 3.20995089512368863973183995384, 3.38348288576375195770306636875, 4.00086460011873360278131875089, 4.04400906574502903625827291092, 4.59525776976627305994409436949, 4.84037565255270880140258086995, 5.06531494702112833482969131234, 5.07248697627071689365880182928, 5.78849425932937401148045538510, 5.85771379787247665333969967904, 6.19744656518248126120863374994, 6.50279266457793012599982116282, 6.69660541094886013523918124332, 7.19068697584096815569796255976, 7.26209424892157839975387495577, 7.55683862445523839866638530289, 7.60486624881837416323507595250, 8.282238517848677303279927684650
