| L(s) = 1 | + 2-s − 3·5-s + 2·8-s − 3·10-s − 3·11-s − 2·13-s + 3·16-s + 2·17-s + 8·19-s − 3·22-s + 6·25-s − 2·26-s + 10·29-s + 8·31-s + 3·32-s + 2·34-s − 6·37-s + 8·38-s − 6·40-s + 14·41-s + 4·43-s + 8·47-s − 5·49-s + 6·50-s + 6·53-s + 9·55-s + 10·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.34·5-s + 0.707·8-s − 0.948·10-s − 0.904·11-s − 0.554·13-s + 3/4·16-s + 0.485·17-s + 1.83·19-s − 0.639·22-s + 6/5·25-s − 0.392·26-s + 1.85·29-s + 1.43·31-s + 0.530·32-s + 0.342·34-s − 0.986·37-s + 1.29·38-s − 0.948·40-s + 2.18·41-s + 0.609·43-s + 1.16·47-s − 5/7·49-s + 0.848·50-s + 0.824·53-s + 1.21·55-s + 1.31·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.532411377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.532411377\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 11 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 - T + T^{2} - 3 T^{3} + p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 5 T^{2} + 16 T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 13 | $D_{6}$ | \( 1 + 2 T + 27 T^{2} + 44 T^{3} + 27 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 2 T - T^{2} + 116 T^{3} - p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 8 T + 41 T^{2} - 144 T^{3} + 41 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 5 T^{2} + 128 T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 10 T + 99 T^{2} - 540 T^{3} + 99 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 8 T + 61 T^{2} - 368 T^{3} + 61 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 41 | $S_4\times C_2$ | \( 1 - 14 T + 167 T^{2} - 1156 T^{3} + 167 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 4 T + 49 T^{2} + 56 T^{3} + 49 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 8 T + 109 T^{2} - 624 T^{3} + 109 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 6 T + 107 T^{2} - 644 T^{3} + 107 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 12 T + 161 T^{2} + 1096 T^{3} + 161 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 6 T + 131 T^{2} + 484 T^{3} + 131 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 4 T + 153 T^{2} + 472 T^{3} + 153 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 8 T + 181 T^{2} + 1008 T^{3} + 181 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 14 T + 223 T^{2} + 1700 T^{3} + 223 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 12 T + 173 T^{2} - 1096 T^{3} + 173 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 129 T^{2} - 16 T^{3} + 129 p T^{4} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 10 T + 215 T^{2} - 1580 T^{3} + 215 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 22 T + 399 T^{2} - 4276 T^{3} + 399 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−9.980772198476117531021630874083, −9.363204275902770412951709587032, −9.211211134832274959655100911751, −8.892852829716815967472081440437, −8.334995642127197229001796897977, −8.263576823772959956278566163982, −7.64264034044433460891990382936, −7.60797619637323739457234157065, −7.54790623810259095050203124969, −7.17284709717532680722351456286, −6.68799032636082666136556134771, −6.15284567326369029615002521121, −6.03710807618258051382479053155, −5.53591530989043077887351693676, −5.00008521939997882539644857419, −4.95519834896995213185497773440, −4.44195366979632959762428238726, −4.40069319073131289028538781247, −3.90729506242122156614587983536, −3.18077298972412545975648584032, −3.14335350508452792539493905225, −2.75720967364800528760863051476, −2.13085287600664917113956499677, −1.15029538288211815341795720771, −0.75317023467176953215891856038,
0.75317023467176953215891856038, 1.15029538288211815341795720771, 2.13085287600664917113956499677, 2.75720967364800528760863051476, 3.14335350508452792539493905225, 3.18077298972412545975648584032, 3.90729506242122156614587983536, 4.40069319073131289028538781247, 4.44195366979632959762428238726, 4.95519834896995213185497773440, 5.00008521939997882539644857419, 5.53591530989043077887351693676, 6.03710807618258051382479053155, 6.15284567326369029615002521121, 6.68799032636082666136556134771, 7.17284709717532680722351456286, 7.54790623810259095050203124969, 7.60797619637323739457234157065, 7.64264034044433460891990382936, 8.263576823772959956278566163982, 8.334995642127197229001796897977, 8.892852829716815967472081440437, 9.211211134832274959655100911751, 9.363204275902770412951709587032, 9.980772198476117531021630874083
