| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{3} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{3} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6172376498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6172376498\) |
| \(L(\frac{5}{2})\) |
|
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
−8.183340382916495792286273973937, −7.75129048586197745267214480398, −7.46274820335089144184036539590, −7.32802857375040382745282652111, −6.88492762520492514702679321174, −6.60897190114826605104654667455, −6.49691726380090003473607106729, −6.00865899900285077822326902680, −5.93554707718714531656307882591, −5.89838162610113166255076279193, −5.41608956515903586054582002292, −5.17836799348553965876181263115, −4.87664844951692625282179973232, −4.22604246466596444307521324181, −4.14243795160563601444384945171, −4.11175205926162876070431757936, −3.30808484270123979424960430841, −3.05681624553291415376237580919, −2.98692777541772668901223241756, −1.99198103166180871729494874080, −1.90744709622484546793163572686, −1.61893608593750226114150517798, −1.09756988350333434088323477837, −0.53649615068815398218941122846, −0.19566998075658332241712031389,
0.19566998075658332241712031389, 0.53649615068815398218941122846, 1.09756988350333434088323477837, 1.61893608593750226114150517798, 1.90744709622484546793163572686, 1.99198103166180871729494874080, 2.98692777541772668901223241756, 3.05681624553291415376237580919, 3.30808484270123979424960430841, 4.11175205926162876070431757936, 4.14243795160563601444384945171, 4.22604246466596444307521324181, 4.87664844951692625282179973232, 5.17836799348553965876181263115, 5.41608956515903586054582002292, 5.89838162610113166255076279193, 5.93554707718714531656307882591, 6.00865899900285077822326902680, 6.49691726380090003473607106729, 6.60897190114826605104654667455, 6.88492762520492514702679321174, 7.32802857375040382745282652111, 7.46274820335089144184036539590, 7.75129048586197745267214480398, 8.183340382916495792286273973937
