| L(s) = 1 | − 3·2-s − 3·3-s + 6·4-s + 3·5-s + 9·6-s − 10·8-s + 6·9-s − 9·10-s − 3·11-s − 18·12-s + 3·13-s − 9·15-s + 15·16-s + 6·17-s − 18·18-s + 6·19-s + 18·20-s + 9·22-s + 30·24-s + 6·25-s − 9·26-s − 10·27-s − 9·29-s + 27·30-s − 15·31-s − 21·32-s + 9·33-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.73·3-s + 3·4-s + 1.34·5-s + 3.67·6-s − 3.53·8-s + 2·9-s − 2.84·10-s − 0.904·11-s − 5.19·12-s + 0.832·13-s − 2.32·15-s + 15/4·16-s + 1.45·17-s − 4.24·18-s + 1.37·19-s + 4.02·20-s + 1.91·22-s + 6.12·24-s + 6/5·25-s − 1.76·26-s − 1.92·27-s − 1.67·29-s + 4.92·30-s − 2.69·31-s − 3.71·32-s + 1.56·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8355254656\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8355254656\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 7 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 - 3 T + 3 T^{2} + 4 T^{3} + 3 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 3 T + 6 T^{2} - 23 T^{3} + 6 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $D_{6}$ | \( 1 - 6 T - 12 T^{2} + 178 T^{3} - 12 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $D_{6}$ | \( 1 - 6 T + 54 T^{2} - 196 T^{3} + 54 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 39 T^{2} - 60 T^{3} + 39 p T^{4} + p^{3} T^{6} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{3} \) |
| 31 | $S_4\times C_2$ | \( 1 + 15 T + 153 T^{2} + 970 T^{3} + 153 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 6 T + 93 T^{2} - 412 T^{3} + 93 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 18 T + 201 T^{2} - 1532 T^{3} + 201 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 12 T + 117 T^{2} + 1016 T^{3} + 117 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 6 T + 78 T^{2} + 352 T^{3} + 78 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 3 T + 42 T^{2} - 359 T^{3} + 42 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 3 T + 150 T^{2} + 305 T^{3} + 150 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 108 T^{2} + 200 T^{3} + 108 p T^{4} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 6 T + 78 T^{2} - 812 T^{3} + 78 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 36 T + 630 T^{2} + 6650 T^{3} + 630 p T^{4} + 36 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 24 T + 381 T^{2} - 3716 T^{3} + 381 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 15 T + 3 p T^{2} - 2270 T^{3} + 3 p^{2} T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 9 T + 141 T^{2} - 1386 T^{3} + 141 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 30 T + 537 T^{2} - 6020 T^{3} + 537 p T^{4} - 30 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 3 T + 219 T^{2} - 748 T^{3} + 219 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.53394327921972488388018782205, −7.37965767220777498487660087177, −7.08282080682791959150386543959, −6.95122759929686426711544858538, −6.40431647299657545263415584140, −6.23232985192461731691614789814, −6.11239495778470289193625384940, −5.78623511154678449843113967644, −5.75667527871166999711037062642, −5.42926046519505342525594085725, −5.06089045538969268194362679472, −4.90464322594887548280334193632, −4.89665493550276033544712120751, −3.94619220557313862472415674070, −3.73730499300540600116237061583, −3.59268125934152572543779077269, −3.21407782880034820566940973282, −2.78981587537140695363190875182, −2.35907171278270428682707484180, −2.14634520294306023495454624702, −1.68183866631211340206989375717, −1.43126271091919644905492186046, −1.20964944454226840211785501760, −0.64780824396376225345205350864, −0.40877548670727699645444367385,
0.40877548670727699645444367385, 0.64780824396376225345205350864, 1.20964944454226840211785501760, 1.43126271091919644905492186046, 1.68183866631211340206989375717, 2.14634520294306023495454624702, 2.35907171278270428682707484180, 2.78981587537140695363190875182, 3.21407782880034820566940973282, 3.59268125934152572543779077269, 3.73730499300540600116237061583, 3.94619220557313862472415674070, 4.89665493550276033544712120751, 4.90464322594887548280334193632, 5.06089045538969268194362679472, 5.42926046519505342525594085725, 5.75667527871166999711037062642, 5.78623511154678449843113967644, 6.11239495778470289193625384940, 6.23232985192461731691614789814, 6.40431647299657545263415584140, 6.95122759929686426711544858538, 7.08282080682791959150386543959, 7.37965767220777498487660087177, 7.53394327921972488388018782205
