| L(s) = 1 | + 15·5-s + 24·7-s − 6·11-s + 48·13-s − 27·17-s + 195·19-s + 27·23-s + 150·25-s + 60·29-s + 279·31-s + 360·35-s − 138·37-s + 66·41-s − 222·43-s + 264·47-s − 345·49-s − 507·53-s − 90·55-s + 960·59-s + 543·61-s + 720·65-s + 1.08e3·67-s + 1.81e3·71-s + 1.36e3·73-s − 144·77-s + 129·79-s + 1.56e3·83-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 1.29·7-s − 0.164·11-s + 1.02·13-s − 0.385·17-s + 2.35·19-s + 0.244·23-s + 6/5·25-s + 0.384·29-s + 1.61·31-s + 1.73·35-s − 0.613·37-s + 0.251·41-s − 0.787·43-s + 0.819·47-s − 1.00·49-s − 1.31·53-s − 0.220·55-s + 2.11·59-s + 1.13·61-s + 1.37·65-s + 1.98·67-s + 3.03·71-s + 2.18·73-s − 0.213·77-s + 0.183·79-s + 2.07·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(20.10977053\) |
| \(L(\frac12)\) |
\(\approx\) |
\(20.10977053\) |
| \(L(\frac{5}{2})\) |
|
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 | $S_4\times C_2$ | \( 1 - 24 T + 921 T^{2} - 12904 T^{3} + 921 p^{3} T^{4} - 24 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 6 T + 513 T^{2} + 60620 T^{3} + 513 p^{3} T^{4} + 6 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 48 T + 3435 T^{2} - 202552 T^{3} + 3435 p^{3} T^{4} - 48 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 27 T - 1158 T^{2} - 163141 T^{3} - 1158 p^{3} T^{4} + 27 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 195 T + 20832 T^{2} - 1720919 T^{3} + 20832 p^{3} T^{4} - 195 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 27 T - 168 T^{2} + 1130141 T^{3} - 168 p^{3} T^{4} - 27 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 60 T + 68067 T^{2} - 2620544 T^{3} + 68067 p^{3} T^{4} - 60 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 9 p T + 111132 T^{2} - 17041763 T^{3} + 111132 p^{3} T^{4} - 9 p^{7} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 138 T + 133923 T^{2} + 14102268 T^{3} + 133923 p^{3} T^{4} + 138 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 66 T + 63627 T^{2} + 6726804 T^{3} + 63627 p^{3} T^{4} - 66 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 222 T + 109929 T^{2} + 9784924 T^{3} + 109929 p^{3} T^{4} + 222 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 264 T + 124605 T^{2} - 55592752 T^{3} + 124605 p^{3} T^{4} - 264 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 507 T + 524166 T^{2} + 154386795 T^{3} + 524166 p^{3} T^{4} + 507 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 960 T + 843525 T^{2} - 409376120 T^{3} + 843525 p^{3} T^{4} - 960 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 543 T + 239946 T^{2} - 10210291 T^{3} + 239946 p^{3} T^{4} - 543 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 1086 T + 604461 T^{2} - 230866972 T^{3} + 604461 p^{3} T^{4} - 1086 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 1818 T + 2146461 T^{2} - 1508229108 T^{3} + 2146461 p^{3} T^{4} - 1818 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 1362 T + 1530147 T^{2} - 1054519260 T^{3} + 1530147 p^{3} T^{4} - 1362 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 129 T + 1152804 T^{2} - 62398365 T^{3} + 1152804 p^{3} T^{4} - 129 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 1569 T + 2366268 T^{2} - 1835088857 T^{3} + 2366268 p^{3} T^{4} - 1569 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 1770 T + 2057019 T^{2} + 1828728060 T^{3} + 2057019 p^{3} T^{4} + 1770 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 336 T + 1258275 T^{2} + 1117216928 T^{3} + 1258275 p^{3} T^{4} + 336 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−7.982482534668081806028546961348, −7.36521872314201080867716142096, −7.20608887066009723392693526474, −6.77847611362638145934252587530, −6.59345867772788218630422796037, −6.37972263326974727102370350970, −6.27567586868156074857304167106, −5.64491202948931594964854409044, −5.41566052438082574440638532111, −5.40352117194220232435691405436, −4.87903667465753922498556029228, −4.87510554808689007287270107032, −4.75296448736718724784634962258, −3.93744080750870509876976615664, −3.82834942765995275982499997319, −3.53469868496814765343216133620, −3.15383637333666551214407826293, −2.76967857049085760224239156433, −2.48610120606485606326511153990, −2.07880550143331065301377415272, −1.77330362164578601760457841603, −1.55807849613900719509430913990, −0.906706082680708087309809639864, −0.811585562832989421100738727383, −0.64153210711821480812153411989,
0.64153210711821480812153411989, 0.811585562832989421100738727383, 0.906706082680708087309809639864, 1.55807849613900719509430913990, 1.77330362164578601760457841603, 2.07880550143331065301377415272, 2.48610120606485606326511153990, 2.76967857049085760224239156433, 3.15383637333666551214407826293, 3.53469868496814765343216133620, 3.82834942765995275982499997319, 3.93744080750870509876976615664, 4.75296448736718724784634962258, 4.87510554808689007287270107032, 4.87903667465753922498556029228, 5.40352117194220232435691405436, 5.41566052438082574440638532111, 5.64491202948931594964854409044, 6.27567586868156074857304167106, 6.37972263326974727102370350970, 6.59345867772788218630422796037, 6.77847611362638145934252587530, 7.20608887066009723392693526474, 7.36521872314201080867716142096, 7.982482534668081806028546961348
