| L(s) = 1 | + 9·3-s − 15·5-s + 21·7-s + 54·9-s + 14·11-s + 20·13-s − 135·15-s − 46·17-s + 74·19-s + 189·21-s + 144·23-s + 150·25-s + 270·27-s + 50·29-s + 182·31-s + 126·33-s − 315·35-s + 30·37-s + 180·39-s − 186·41-s + 716·43-s − 810·45-s + 400·47-s + 294·49-s − 414·51-s − 236·53-s − 210·55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 1.34·5-s + 1.13·7-s + 2·9-s + 0.383·11-s + 0.426·13-s − 2.32·15-s − 0.656·17-s + 0.893·19-s + 1.96·21-s + 1.30·23-s + 6/5·25-s + 1.92·27-s + 0.320·29-s + 1.05·31-s + 0.664·33-s − 1.52·35-s + 0.133·37-s + 0.739·39-s − 0.708·41-s + 2.53·43-s − 2.68·45-s + 1.24·47-s + 6/7·49-s − 1.13·51-s − 0.611·53-s − 0.514·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{3} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{3} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(16.12497501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(16.12497501\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{3} \) |
| 5 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 7 | $C_1$ | \( ( 1 - p T )^{3} \) |
| good | 11 | $S_4\times C_2$ | \( 1 - 14 T + 1361 T^{2} - 74900 T^{3} + 1361 p^{3} T^{4} - 14 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 20 T - 277 T^{2} + 144232 T^{3} - 277 p^{3} T^{4} - 20 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 46 T + 4655 T^{2} + 691108 T^{3} + 4655 p^{3} T^{4} + 46 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 74 T + 19705 T^{2} - 1013724 T^{3} + 19705 p^{3} T^{4} - 74 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 144 T + 11413 T^{2} - 423904 T^{3} + 11413 p^{3} T^{4} - 144 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 50 T + 62603 T^{2} - 2548652 T^{3} + 62603 p^{3} T^{4} - 50 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 182 T + 32981 T^{2} - 711508 T^{3} + 32981 p^{3} T^{4} - 182 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 30 T + 120259 T^{2} - 1065396 T^{3} + 120259 p^{3} T^{4} - 30 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 186 T + 135367 T^{2} + 28849452 T^{3} + 135367 p^{3} T^{4} + 186 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 716 T + 398617 T^{2} - 124472904 T^{3} + 398617 p^{3} T^{4} - 716 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 400 T + 354013 T^{2} - 84391136 T^{3} + 354013 p^{3} T^{4} - 400 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 236 T + 173443 T^{2} + 48465928 T^{3} + 173443 p^{3} T^{4} + 236 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 272 T - 59335 T^{2} + 136740000 T^{3} - 59335 p^{3} T^{4} - 272 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 18 p T + 748939 T^{2} - 377268860 T^{3} + 748939 p^{3} T^{4} - 18 p^{7} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 1044 T + 621265 T^{2} - 260401464 T^{3} + 621265 p^{3} T^{4} - 1044 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 1542 T + 1620573 T^{2} - 1114587156 T^{3} + 1620573 p^{3} T^{4} - 1542 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 996 T + 1237975 T^{2} - 692148824 T^{3} + 1237975 p^{3} T^{4} - 996 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 24 p T + 2605389 T^{2} - 2070950960 T^{3} + 2605389 p^{3} T^{4} - 24 p^{7} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 604 T + 927633 T^{2} - 708146728 T^{3} + 927633 p^{3} T^{4} - 604 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 146 T + 1858247 T^{2} - 181892636 T^{3} + 1858247 p^{3} T^{4} - 146 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 1072 T + 2778015 T^{2} - 1943515424 T^{3} + 2778015 p^{3} T^{4} - 1072 p^{6} T^{5} + p^{9} T^{6} \) |
| 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
−8.684303690995114614583758167137, −8.325576470824751289931563519178, −8.189886784573142897293061522074, −7.86362084719092841615478257312, −7.58339725283266956062421829900, −7.52131800392452181975079388279, −7.10354117694684984286149033275, −6.66255919393149691174727989028, −6.52205463542081927888097921433, −6.33682683385936478172551515629, −5.39375643795821983496445029326, −5.29929938127046261348716541520, −5.05419050902688867216129598953, −4.57982747657367054799417318920, −4.20342733310857758648737637716, −4.10832160399868422257659432756, −3.58095219364929280516047315383, −3.41690052448454187378955877837, −3.13690482562260010706983312277, −2.39160581029532535894898749532, −2.21750860620593127739737408315, −2.09869116471603358382365240857, −0.968578693405646456285507107130, −0.882544966930941007490020085849, −0.800402496381955124981188393598,
0.800402496381955124981188393598, 0.882544966930941007490020085849, 0.968578693405646456285507107130, 2.09869116471603358382365240857, 2.21750860620593127739737408315, 2.39160581029532535894898749532, 3.13690482562260010706983312277, 3.41690052448454187378955877837, 3.58095219364929280516047315383, 4.10832160399868422257659432756, 4.20342733310857758648737637716, 4.57982747657367054799417318920, 5.05419050902688867216129598953, 5.29929938127046261348716541520, 5.39375643795821983496445029326, 6.33682683385936478172551515629, 6.52205463542081927888097921433, 6.66255919393149691174727989028, 7.10354117694684984286149033275, 7.52131800392452181975079388279, 7.58339725283266956062421829900, 7.86362084719092841615478257312, 8.189886784573142897293061522074, 8.325576470824751289931563519178, 8.684303690995114614583758167137
