| L(s) = 1 | − 3·2-s − 3·3-s + 6·4-s − 6·5-s + 9·6-s − 3·7-s − 10·8-s + 6·9-s + 18·10-s − 12·11-s − 18·12-s + 9·14-s + 18·15-s + 15·16-s − 12·17-s − 18·18-s − 36·20-s + 9·21-s + 36·22-s − 12·23-s + 30·24-s + 12·25-s − 10·27-s − 18·28-s + 6·29-s − 54·30-s + 3·31-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.73·3-s + 3·4-s − 2.68·5-s + 3.67·6-s − 1.13·7-s − 3.53·8-s + 2·9-s + 5.69·10-s − 3.61·11-s − 5.19·12-s + 2.40·14-s + 4.64·15-s + 15/4·16-s − 2.91·17-s − 4.24·18-s − 8.04·20-s + 1.96·21-s + 7.67·22-s − 2.50·23-s + 6.12·24-s + 12/5·25-s − 1.92·27-s − 3.40·28-s + 1.11·29-s − 9.85·30-s + 0.538·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.01801462462\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01801462462\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 19 | | \( 1 \) |
| good | 5 | $A_4\times C_2$ | \( 1 + 6 T + 24 T^{2} + 63 T^{3} + 24 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + 3 T + 15 T^{2} + 43 T^{3} + 15 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 12 T + 78 T^{2} + 315 T^{3} + 78 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $C_6$ | \( 1 - 19 T^{3} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 12 T + 96 T^{2} + 27 p T^{3} + 96 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 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 | $A_4\times C_2$ | \( 1 - 6 T + 60 T^{2} - 297 T^{3} + 60 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 3 T + 87 T^{2} - 169 T^{3} + 87 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 3 T + 102 T^{2} + 203 T^{3} + 102 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 9 T + 3 p T^{2} - 711 T^{3} + 3 p^{2} T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 3 T + 105 T^{2} + 259 T^{3} + 105 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 15 T + 177 T^{2} + 1251 T^{3} + 177 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 3 T + 78 T^{2} - 99 T^{3} + 78 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 60 T^{2} - 153 T^{3} + 60 p T^{4} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 3 T + 159 T^{2} + 367 T^{3} + 159 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 6 T + 174 T^{2} - 715 T^{3} + 174 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 9 T + 3 p T^{2} - 1197 T^{3} + 3 p^{2} T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 3 T - 15 T^{2} + 301 T^{3} - 15 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 9 T + 189 T^{2} + 1349 T^{3} + 189 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 3 T - 21 T^{2} + 1395 T^{3} - 21 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 3 T - 3 T^{2} - 1359 T^{3} - 3 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 12 T + 168 T^{2} - 1051 T^{3} + 168 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.153536459252923568363488940050, −7.78174084811435678533601274918, −7.66921702353050285878676493469, −7.50921123301525814017976239660, −6.94293886081660087225906776248, −6.81970406907095281441253216077, −6.77893836226605670961602814473, −6.14305302619220385256862163190, −6.08425040842621976481591711199, −6.04101984839712838059217422723, −5.24141605467512553209939110722, −5.21985410411543365466991427871, −4.87630038812921414582948533016, −4.32477722474393997363537319111, −4.23414418535109839265860834718, −4.13605496995761638505157952239, −3.27216467772833697452763983271, −3.25493163809521606105904737843, −2.92111680654933531969673815537, −2.30239813908795665656746712681, −1.98510688825886346194303371566, −1.94643732661164759335011404428, −0.60975817559061267230126568859, −0.41579777779247089120206764868, −0.19048034515854953435294516856,
0.19048034515854953435294516856, 0.41579777779247089120206764868, 0.60975817559061267230126568859, 1.94643732661164759335011404428, 1.98510688825886346194303371566, 2.30239813908795665656746712681, 2.92111680654933531969673815537, 3.25493163809521606105904737843, 3.27216467772833697452763983271, 4.13605496995761638505157952239, 4.23414418535109839265860834718, 4.32477722474393997363537319111, 4.87630038812921414582948533016, 5.21985410411543365466991427871, 5.24141605467512553209939110722, 6.04101984839712838059217422723, 6.08425040842621976481591711199, 6.14305302619220385256862163190, 6.77893836226605670961602814473, 6.81970406907095281441253216077, 6.94293886081660087225906776248, 7.50921123301525814017976239660, 7.66921702353050285878676493469, 7.78174084811435678533601274918, 8.153536459252923568363488940050
