| L(s) = 1 | − 2·3-s − 2·7-s − 2·9-s + 11-s + 13-s + 2·17-s + 3·19-s + 4·21-s − 8·23-s + 5·27-s − 7·29-s − 11·31-s − 2·33-s − 5·37-s − 2·39-s + 13·41-s − 11·43-s − 19·47-s − 14·49-s − 4·51-s + 11·53-s − 6·57-s + 6·59-s + 7·61-s + 4·63-s − 3·67-s + 16·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.755·7-s − 2/3·9-s + 0.301·11-s + 0.277·13-s + 0.485·17-s + 0.688·19-s + 0.872·21-s − 1.66·23-s + 0.962·27-s − 1.29·29-s − 1.97·31-s − 0.348·33-s − 0.821·37-s − 0.320·39-s + 2.03·41-s − 1.67·43-s − 2.77·47-s − 2·49-s − 0.560·51-s + 1.51·53-s − 0.794·57-s + 0.781·59-s + 0.896·61-s + 0.503·63-s − 0.366·67-s + 1.92·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 2 T + 2 p T^{2} + 11 T^{3} + 2 p^{2} T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 2 T + 18 T^{2} + 27 T^{3} + 18 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - T + 7 T^{2} - 37 T^{3} + 7 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - T + 31 T^{2} - 23 T^{3} + 31 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 2 T + 48 T^{2} - 67 T^{3} + 48 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 8 T + 66 T^{2} + 359 T^{3} + 66 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 7 T + 77 T^{2} + 381 T^{3} + 77 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 11 T + 87 T^{2} + 441 T^{3} + 87 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 5 T + 39 T^{2} - 35 T^{3} + 39 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 13 T + 159 T^{2} - 1091 T^{3} + 159 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 11 T + 152 T^{2} + 919 T^{3} + 152 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 19 T + 209 T^{2} + 1723 T^{3} + 209 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 11 T + 182 T^{2} - 1139 T^{3} + 182 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 6 T + T^{2} + 148 T^{3} + p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 7 T + 153 T^{2} - 879 T^{3} + 153 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 3 T + 73 T^{2} - 197 T^{3} + 73 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 5 T + 140 T^{2} - 833 T^{3} + 140 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 9 T + 201 T^{2} - 1287 T^{3} + 201 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 18 T + 238 T^{2} - 2237 T^{3} + 238 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 3 T + T^{2} - 251 T^{3} + p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 84 T^{2} - 857 T^{3} + 84 p T^{4} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 3 T + 219 T^{2} + 501 T^{3} + 219 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.29425327790874561734963007577, −6.87645300920565498585604899299, −6.75517790451181659788438233620, −6.69049911760215726282006542939, −6.15903779026175051130915760813, −6.15800730636925364536696055017, −6.03361105723144274472905109764, −5.52241383367963571635531911415, −5.51085922211621842519468572854, −5.32249545051676188153754877809, −5.11168155085938230775690152433, −4.81135980754431240052659239051, −4.56479500316353670185119705896, −3.98716838358164976359594551708, −3.90615212103289621651175021774, −3.65550151633669366342639790633, −3.42663000976611058933106669918, −3.28890305703083853239876872728, −2.96458914110625023221374534514, −2.47996320099660531125103999457, −2.12330802835357176573579754524, −1.98582757715842949354529472289, −1.66005377537821495875329695912, −1.14803887850151433699863163189, −0.876334310857851702427054702958, 0, 0, 0,
0.876334310857851702427054702958, 1.14803887850151433699863163189, 1.66005377537821495875329695912, 1.98582757715842949354529472289, 2.12330802835357176573579754524, 2.47996320099660531125103999457, 2.96458914110625023221374534514, 3.28890305703083853239876872728, 3.42663000976611058933106669918, 3.65550151633669366342639790633, 3.90615212103289621651175021774, 3.98716838358164976359594551708, 4.56479500316353670185119705896, 4.81135980754431240052659239051, 5.11168155085938230775690152433, 5.32249545051676188153754877809, 5.51085922211621842519468572854, 5.52241383367963571635531911415, 6.03361105723144274472905109764, 6.15800730636925364536696055017, 6.15903779026175051130915760813, 6.69049911760215726282006542939, 6.75517790451181659788438233620, 6.87645300920565498585604899299, 7.29425327790874561734963007577
