| L(s) = 1 | + 3·5-s − 3·7-s − 6·11-s − 3·13-s + 6·17-s + 3·19-s − 12·23-s − 6·25-s + 9·29-s + 3·31-s − 9·35-s − 3·37-s + 3·43-s − 3·47-s + 6·49-s + 6·53-s − 18·55-s + 3·59-s + 6·61-s − 9·65-s + 12·67-s − 9·71-s − 21·73-s + 18·77-s + 21·79-s + 18·83-s + 18·85-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 1.13·7-s − 1.80·11-s − 0.832·13-s + 1.45·17-s + 0.688·19-s − 2.50·23-s − 6/5·25-s + 1.67·29-s + 0.538·31-s − 1.52·35-s − 0.493·37-s + 0.457·43-s − 0.437·47-s + 6/7·49-s + 0.824·53-s − 2.42·55-s + 0.390·59-s + 0.768·61-s − 1.11·65-s + 1.46·67-s − 1.06·71-s − 2.45·73-s + 2.05·77-s + 2.36·79-s + 1.97·83-s + 1.95·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.952060539\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.952060539\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 5 | $A_4\times C_2$ | \( 1 - 3 T + 3 p T^{2} - 27 T^{3} + 3 p^{2} T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 6 T + 42 T^{2} + 135 T^{3} + 42 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 + 3 T + 6 T^{2} - 29 T^{3} + 6 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 - 6 T + 60 T^{2} - 207 T^{3} + 60 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 - 3 T + 51 T^{2} - 97 T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 12 T + 96 T^{2} + 549 T^{3} + 96 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - 9 T + 51 T^{2} - 189 T^{3} + 51 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 3 T + 15 T^{2} + 137 T^{3} + 15 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 3 T + 33 T^{2} - 101 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 114 T^{2} - 9 T^{3} + 114 p T^{4} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 3 T + 123 T^{2} - 259 T^{3} + 123 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 3 T + 87 T^{2} + 333 T^{3} + 87 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 6 T + 150 T^{2} - 639 T^{3} + 150 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 3 T + 105 T^{2} - 405 T^{3} + 105 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 - 6 T + 168 T^{2} - 713 T^{3} + 168 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 12 T + 222 T^{2} - 1591 T^{3} + 222 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 9 T + 159 T^{2} + 1305 T^{3} + 159 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + 21 T + 303 T^{2} + 2797 T^{3} + 303 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 21 T + 357 T^{2} - 3499 T^{3} + 357 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 18 T + 294 T^{2} - 2997 T^{3} + 294 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 12 T + 204 T^{2} - 1323 T^{3} + 204 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 3 T + 123 T^{2} + 259 T^{3} + 123 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−6.80687196623650056496650641524, −6.35246626872389726915041649498, −6.34397593266677109621226637690, −6.24589951324699001683968805853, −5.72646314981267686523808582316, −5.72101095716759272766771176287, −5.60819761697063157258232597282, −5.17420839596661862399340767630, −5.08021543544621423272514686227, −4.99552326476754747711043674059, −4.24705754304083014948546664196, −4.22733950306204799767435048654, −4.19135645036486162806437656035, −3.43086501600051465782025140493, −3.42627479720818026947894276282, −3.31512700416462232078905957249, −2.71816713077920680324747703884, −2.59276166639788643258080061202, −2.47999482039665947338332982835, −1.94779109086205973353943929018, −1.93539549724074097578053486565, −1.60837501730279034657221610603, −0.927342309117779034536445153365, −0.59449773905933174064225209636, −0.34094658125082594052727031676,
0.34094658125082594052727031676, 0.59449773905933174064225209636, 0.927342309117779034536445153365, 1.60837501730279034657221610603, 1.93539549724074097578053486565, 1.94779109086205973353943929018, 2.47999482039665947338332982835, 2.59276166639788643258080061202, 2.71816713077920680324747703884, 3.31512700416462232078905957249, 3.42627479720818026947894276282, 3.43086501600051465782025140493, 4.19135645036486162806437656035, 4.22733950306204799767435048654, 4.24705754304083014948546664196, 4.99552326476754747711043674059, 5.08021543544621423272514686227, 5.17420839596661862399340767630, 5.60819761697063157258232597282, 5.72101095716759272766771176287, 5.72646314981267686523808582316, 6.24589951324699001683968805853, 6.34397593266677109621226637690, 6.35246626872389726915041649498, 6.80687196623650056496650641524
