| L(s) = 1 | − 9·3-s − 10·5-s + 21·7-s + 54·9-s − 50·13-s + 90·15-s + 30·17-s − 140·19-s − 189·21-s − 56·23-s + 25·25-s − 270·27-s + 298·29-s + 80·31-s − 210·35-s + 10·37-s + 450·39-s + 390·41-s − 784·43-s − 540·45-s − 248·47-s + 294·49-s − 270·51-s + 10·53-s + 1.26e3·57-s − 1.50e3·59-s − 810·61-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.894·5-s + 1.13·7-s + 2·9-s − 1.06·13-s + 1.54·15-s + 0.428·17-s − 1.69·19-s − 1.96·21-s − 0.507·23-s + 1/5·25-s − 1.92·27-s + 1.90·29-s + 0.463·31-s − 1.01·35-s + 0.0444·37-s + 1.84·39-s + 1.48·41-s − 2.78·43-s − 1.78·45-s − 0.769·47-s + 6/7·49-s − 0.741·51-s + 0.0259·53-s + 2.92·57-s − 3.30·59-s − 1.70·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{3} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{3} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 7 | $C_1$ | \( ( 1 - p T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 + 2 p T + 3 p^{2} T^{2} + 44 p^{2} T^{3} + 3 p^{5} T^{4} + 2 p^{7} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 1153 T^{2} - 45280 T^{3} + 1153 p^{3} T^{4} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 50 T + 5611 T^{2} + 192780 T^{3} + 5611 p^{3} T^{4} + 50 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 30 T + 8359 T^{2} - 18900 T^{3} + 8359 p^{3} T^{4} - 30 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 140 T + 17457 T^{2} + 101400 p T^{3} + 17457 p^{3} T^{4} + 140 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 56 T + 1213 T^{2} - 1923088 T^{3} + 1213 p^{3} T^{4} + 56 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 298 T + 66235 T^{2} - 9210140 T^{3} + 66235 p^{3} T^{4} - 298 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 80 T + 19773 T^{2} + 3933600 T^{3} + 19773 p^{3} T^{4} - 80 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 10 T + 97699 T^{2} + 3965700 T^{3} + 97699 p^{3} T^{4} - 10 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 390 T + 157663 T^{2} - 32399780 T^{3} + 157663 p^{3} T^{4} - 390 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 784 T + 422073 T^{2} + 137106528 T^{3} + 422073 p^{3} T^{4} + 784 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 248 T + 330637 T^{2} + 51948304 T^{3} + 330637 p^{3} T^{4} + 248 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 10 T + 129971 T^{2} + 29769540 T^{3} + 129971 p^{3} T^{4} - 10 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 1500 T + 1276057 T^{2} + 706438120 T^{3} + 1276057 p^{3} T^{4} + 1500 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 810 T + 308443 T^{2} + 91697820 T^{3} + 308443 p^{3} T^{4} + 810 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 1272 T + 1429617 T^{2} + 836207696 T^{3} + 1429617 p^{3} T^{4} + 1272 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 160 T + 165933 T^{2} + 373241120 T^{3} + 165933 p^{3} T^{4} + 160 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 1170 T + 1041751 T^{2} + 572317980 T^{3} + 1041751 p^{3} T^{4} + 1170 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 840 T + 1558957 T^{2} + 798709360 T^{3} + 1558957 p^{3} T^{4} + 840 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 1564 T + 2323793 T^{2} + 1821572008 T^{3} + 2323793 p^{3} T^{4} + 1564 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 2 p T + 1900335 T^{2} + 197453740 T^{3} + 1900335 p^{3} T^{4} + 2 p^{7} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 130 T + 12687 p T^{2} - 162952740 T^{3} + 12687 p^{4} T^{4} + 130 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−9.760509852026504197787743767524, −8.827890134616150933719200505625, −8.789840473826226233174162567925, −8.643228709432220855759833117510, −8.047187457760685254166696115718, −7.80456620525664472118248195230, −7.73155016536053322978090866238, −7.22570346622461860010849368374, −7.15634855448456680064702819070, −6.52851178757177014121384205683, −6.29623731785609564667519755091, −6.20773714306868944398740896966, −5.83179222014330723431128134193, −5.25049722583240322900079994557, −5.08386715240211384320971480935, −4.74004322517923287158694997435, −4.39315571552407854741480503536, −4.24306023014226820316252191264, −4.12646327211615724004941601515, −3.16066993648427611766549310148, −2.94801063821003400999808686405, −2.52456747001431324889038241534, −1.72446616297686871557444133392, −1.39977699927559998251535100701, −1.22315406409809841264414827413, 0, 0, 0,
1.22315406409809841264414827413, 1.39977699927559998251535100701, 1.72446616297686871557444133392, 2.52456747001431324889038241534, 2.94801063821003400999808686405, 3.16066993648427611766549310148, 4.12646327211615724004941601515, 4.24306023014226820316252191264, 4.39315571552407854741480503536, 4.74004322517923287158694997435, 5.08386715240211384320971480935, 5.25049722583240322900079994557, 5.83179222014330723431128134193, 6.20773714306868944398740896966, 6.29623731785609564667519755091, 6.52851178757177014121384205683, 7.15634855448456680064702819070, 7.22570346622461860010849368374, 7.73155016536053322978090866238, 7.80456620525664472118248195230, 8.047187457760685254166696115718, 8.643228709432220855759833117510, 8.789840473826226233174162567925, 8.827890134616150933719200505625, 9.760509852026504197787743767524
