| L(s) = 1 | + 3·3-s − 5-s + 3·7-s + 6·9-s − 2·11-s + 3·13-s − 3·15-s − 8·17-s − 4·19-s + 9·21-s − 3·23-s − 9·25-s + 10·27-s − 12·29-s − 4·31-s − 6·33-s − 3·35-s + 2·37-s + 9·39-s − 16·41-s + 3·43-s − 6·45-s − 18·47-s + 6·49-s − 24·51-s − 11·53-s + 2·55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.447·5-s + 1.13·7-s + 2·9-s − 0.603·11-s + 0.832·13-s − 0.774·15-s − 1.94·17-s − 0.917·19-s + 1.96·21-s − 0.625·23-s − 9/5·25-s + 1.92·27-s − 2.22·29-s − 0.718·31-s − 1.04·33-s − 0.507·35-s + 0.328·37-s + 1.44·39-s − 2.49·41-s + 0.457·43-s − 0.894·45-s − 2.62·47-s + 6/7·49-s − 3.36·51-s − 1.51·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 7^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 7^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| 23 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 + T + 2 p T^{2} + 12 T^{3} + 2 p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 2 T + 2 p T^{2} + 48 T^{3} + 2 p^{2} T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 3 T + 38 T^{2} - 76 T^{3} + 38 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 8 T + 60 T^{2} + 270 T^{3} + 60 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 4 T + 12 T^{2} - 44 T^{3} + 12 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 12 T + 92 T^{2} + 482 T^{3} + 92 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 4 T + 92 T^{2} + 240 T^{3} + 92 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 2 T + 22 T^{2} - 206 T^{3} + 22 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 16 T + 176 T^{2} + 1238 T^{3} + 176 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 3 T + 128 T^{2} - 254 T^{3} + 128 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 18 T + 233 T^{2} + 1820 T^{3} + 233 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 11 T + 194 T^{2} + 1192 T^{3} + 194 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 19 T + 240 T^{2} + 2046 T^{3} + 240 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 9 T + 36 T^{2} - 88 T^{3} + 36 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 3 T + 122 T^{2} + 214 T^{3} + 122 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 9 T - 22 T^{2} + 458 T^{3} - 22 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 8 T + 120 T^{2} - 1274 T^{3} + 120 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 18 T + 302 T^{2} - 2908 T^{3} + 302 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 4 T + 164 T^{2} + 428 T^{3} + 164 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 3 T + 56 T^{2} + 920 T^{3} + 56 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 16 T + 256 T^{2} + 2206 T^{3} + 256 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.67667733572195918036227013610, −6.89231160967650542736141320201, −6.85897995574540721311179824843, −6.84395454977280152928313257696, −6.37886776184546405405994371867, −6.14581331503310050383597523245, −6.14064774303998535772603750802, −5.50379329424009880217617469404, −5.25000249133990445431520371047, −5.24344766672783959241678497733, −4.92234037204368515572225867887, −4.44881142292906031683589755059, −4.34586912968767802836295442907, −4.17124867134059138436755191754, −3.89414259404861715371056479602, −3.72799314032337939568877598408, −3.32297434628509511644915970312, −3.13786577984137574191803727846, −2.99906996575158976123689015467, −2.26853189666214416652210335822, −2.21122049168949333093031203549, −2.09120734402972391108175203718, −1.63345868575511824481165799233, −1.52748723371001040326515311827, −1.29960953490868288320184326637, 0, 0, 0,
1.29960953490868288320184326637, 1.52748723371001040326515311827, 1.63345868575511824481165799233, 2.09120734402972391108175203718, 2.21122049168949333093031203549, 2.26853189666214416652210335822, 2.99906996575158976123689015467, 3.13786577984137574191803727846, 3.32297434628509511644915970312, 3.72799314032337939568877598408, 3.89414259404861715371056479602, 4.17124867134059138436755191754, 4.34586912968767802836295442907, 4.44881142292906031683589755059, 4.92234037204368515572225867887, 5.24344766672783959241678497733, 5.25000249133990445431520371047, 5.50379329424009880217617469404, 6.14064774303998535772603750802, 6.14581331503310050383597523245, 6.37886776184546405405994371867, 6.84395454977280152928313257696, 6.85897995574540721311179824843, 6.89231160967650542736141320201, 7.67667733572195918036227013610
