| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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
−8.551461093160894423181193566992, −8.152371145571219894667841733405, −7.994359911729091262771557118885, −7.88375002073260156012949291420, −7.71812629623321670961146813273, −7.10835740305835903963143590385, −6.87381590083381052954622797379, −6.48194885747403117978648176427, −6.37252904337519794972227129130, −6.24324375058328167995053844109, −5.93443749351693622668376734064, −5.62727606246215731571720747450, −5.53337862469520074919136207588, −4.78877237873351340495902590237, −4.78208392998233713160815820697, −4.76841061830974078263335124269, −4.08981849159772921019522733733, −3.77681515280349990826022879062, −3.65467397697074916693274332468, −2.84553871752581651325104418607, −2.83998976776324888751182321096, −2.71064648842484936641733728989, −1.95437803332624454046645372862, −1.63498848647341875565445815630, −1.20811503734359560501994971125, 0, 0, 0,
1.20811503734359560501994971125, 1.63498848647341875565445815630, 1.95437803332624454046645372862, 2.71064648842484936641733728989, 2.83998976776324888751182321096, 2.84553871752581651325104418607, 3.65467397697074916693274332468, 3.77681515280349990826022879062, 4.08981849159772921019522733733, 4.76841061830974078263335124269, 4.78208392998233713160815820697, 4.78877237873351340495902590237, 5.53337862469520074919136207588, 5.62727606246215731571720747450, 5.93443749351693622668376734064, 6.24324375058328167995053844109, 6.37252904337519794972227129130, 6.48194885747403117978648176427, 6.87381590083381052954622797379, 7.10835740305835903963143590385, 7.71812629623321670961146813273, 7.88375002073260156012949291420, 7.994359911729091262771557118885, 8.152371145571219894667841733405, 8.551461093160894423181193566992
