| L(s) = 1 | − 15·5-s − 14·7-s + 22·11-s + 8·13-s − 34·17-s + 4·19-s + 176·23-s + 150·25-s + 98·29-s − 88·31-s + 210·35-s + 284·37-s − 8·41-s − 504·43-s + 280·47-s − 621·49-s − 150·53-s − 330·55-s + 350·59-s + 350·61-s − 120·65-s − 804·67-s + 500·71-s − 486·73-s − 308·77-s − 1.59e3·79-s − 684·83-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.755·7-s + 0.603·11-s + 0.170·13-s − 0.485·17-s + 0.0482·19-s + 1.59·23-s + 6/5·25-s + 0.627·29-s − 0.509·31-s + 1.01·35-s + 1.26·37-s − 0.0304·41-s − 1.78·43-s + 0.868·47-s − 1.81·49-s − 0.388·53-s − 0.809·55-s + 0.772·59-s + 0.734·61-s − 0.228·65-s − 1.46·67-s + 0.835·71-s − 0.779·73-s − 0.455·77-s − 2.26·79-s − 0.904·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{6} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{6} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 + 2 p T + 817 T^{2} + 9196 T^{3} + 817 p^{3} T^{4} + 2 p^{7} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 2 p T + 2149 T^{2} - 69196 T^{3} + 2149 p^{3} T^{4} - 2 p^{7} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 8 T + 2127 T^{2} - 119632 T^{3} + 2127 p^{3} T^{4} - 8 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 2 p T + 7951 T^{2} + 430972 T^{3} + 7951 p^{3} T^{4} + 2 p^{7} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 4 T + 7649 T^{2} - 529240 T^{3} + 7649 p^{3} T^{4} - 4 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 176 T + 35205 T^{2} - 4252064 T^{3} + 35205 p^{3} T^{4} - 176 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 98 T + 24635 T^{2} - 7058028 T^{3} + 24635 p^{3} T^{4} - 98 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 88 T + 52685 T^{2} + 3535312 T^{3} + 52685 p^{3} T^{4} + 88 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 284 T + 49559 T^{2} - 13028696 T^{3} + 49559 p^{3} T^{4} - 284 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 8 T + 121451 T^{2} + 10434960 T^{3} + 121451 p^{3} T^{4} + 8 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 504 T + 174009 T^{2} + 40942288 T^{3} + 174009 p^{3} T^{4} + 504 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 280 T + 115997 T^{2} - 54406224 T^{3} + 115997 p^{3} T^{4} - 280 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 150 T + 253683 T^{2} + 44718980 T^{3} + 253683 p^{3} T^{4} + 150 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 350 T + 202005 T^{2} + 17161444 T^{3} + 202005 p^{3} T^{4} - 350 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 350 T + 204635 T^{2} - 134954132 T^{3} + 204635 p^{3} T^{4} - 350 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 12 p T + 1077441 T^{2} + 489158232 T^{3} + 1077441 p^{3} T^{4} + 12 p^{7} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 500 T + 1072245 T^{2} - 353953688 T^{3} + 1072245 p^{3} T^{4} - 500 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 486 T + 1084311 T^{2} + 377044724 T^{3} + 1084311 p^{3} T^{4} + 486 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 1592 T + 2161789 T^{2} + 1645964560 T^{3} + 2161789 p^{3} T^{4} + 1592 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 684 T + 1381713 T^{2} + 561532168 T^{3} + 1381713 p^{3} T^{4} + 684 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 668 T + 2135307 T^{2} + 937072184 T^{3} + 2135307 p^{3} T^{4} + 668 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 1394 T + 2868623 T^{2} + 2439375164 T^{3} + 2868623 p^{3} T^{4} + 1394 p^{6} T^{5} + p^{9} T^{6} \) |
| 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
−8.676445533957566362960955340327, −8.144984298150885386867926988664, −7.953051590373126438233373960666, −7.82043602654827587803491665629, −7.30092025440511508407231959742, −7.14753993890884525165337900317, −6.85514750001079727447740080250, −6.65854241764556712109331020424, −6.40511679546833741412963600861, −6.15906081975840248381496914027, −5.74295469791199587196799942406, −5.31421099950500302813406276346, −5.08768026533605371090605198207, −4.85289542193259311843993670900, −4.37780636481826741848319874019, −4.23831983983188145751446678522, −3.79927882848732593860943729602, −3.62125277674729755704514004242, −3.32406267100973751941857449855, −2.75637228871461436459842038280, −2.69775832656157036340378562775, −2.40708535085167749803239582458, −1.37788600027133611220236036470, −1.22956791930666257803616353082, −1.19570036774966284228211679763, 0, 0, 0,
1.19570036774966284228211679763, 1.22956791930666257803616353082, 1.37788600027133611220236036470, 2.40708535085167749803239582458, 2.69775832656157036340378562775, 2.75637228871461436459842038280, 3.32406267100973751941857449855, 3.62125277674729755704514004242, 3.79927882848732593860943729602, 4.23831983983188145751446678522, 4.37780636481826741848319874019, 4.85289542193259311843993670900, 5.08768026533605371090605198207, 5.31421099950500302813406276346, 5.74295469791199587196799942406, 6.15906081975840248381496914027, 6.40511679546833741412963600861, 6.65854241764556712109331020424, 6.85514750001079727447740080250, 7.14753993890884525165337900317, 7.30092025440511508407231959742, 7.82043602654827587803491665629, 7.953051590373126438233373960666, 8.144984298150885386867926988664, 8.676445533957566362960955340327
