| L(s) = 1 | − 15·5-s + 10·7-s + 28·11-s − 78·13-s − 11·17-s + 71·19-s − 25·23-s + 150·25-s + 118·29-s + 107·31-s − 150·35-s − 410·37-s + 592·41-s − 52·43-s − 580·47-s − 225·49-s + 169·53-s − 420·55-s + 234·59-s − 673·61-s + 1.17e3·65-s − 386·67-s + 16·71-s − 892·73-s + 280·77-s − 1.26e3·79-s − 1.81e3·83-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.539·7-s + 0.767·11-s − 1.66·13-s − 0.156·17-s + 0.857·19-s − 0.226·23-s + 6/5·25-s + 0.755·29-s + 0.619·31-s − 0.724·35-s − 1.82·37-s + 2.25·41-s − 0.184·43-s − 1.80·47-s − 0.655·49-s + 0.437·53-s − 1.02·55-s + 0.516·59-s − 1.41·61-s + 2.23·65-s − 0.703·67-s + 0.0267·71-s − 1.43·73-s + 0.414·77-s − 1.79·79-s − 2.40·83-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $S_4\times C_2$ | \( 1 - 10 T + 325 T^{2} - 4052 T^{3} + 325 p^{3} T^{4} - 10 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 28 T + 111 p T^{2} - 66112 T^{3} + 111 p^{4} T^{4} - 28 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 6 p T + 579 p T^{2} + 336036 T^{3} + 579 p^{4} T^{4} + 6 p^{7} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 11 T + 9706 T^{2} - 12229 T^{3} + 9706 p^{3} T^{4} + 11 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 71 T + 19688 T^{2} - 915683 T^{3} + 19688 p^{3} T^{4} - 71 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 25 T + 2176 T^{2} - 755303 T^{3} + 2176 p^{3} T^{4} + 25 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 118 T + 15375 T^{2} - 3162436 T^{3} + 15375 p^{3} T^{4} - 118 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 107 T + 26244 T^{2} - 9258271 T^{3} + 26244 p^{3} T^{4} - 107 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 410 T + 126819 T^{2} + 26481628 T^{3} + 126819 p^{3} T^{4} + 410 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 592 T + 301279 T^{2} - 2091704 p T^{3} + 301279 p^{3} T^{4} - 592 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 52 T + 175069 T^{2} + 917504 T^{3} + 175069 p^{3} T^{4} + 52 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 580 T + 309773 T^{2} + 92211384 T^{3} + 309773 p^{3} T^{4} + 580 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 169 T + 419518 T^{2} - 50902417 T^{3} + 419518 p^{3} T^{4} - 169 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 234 T + 385665 T^{2} - 30035844 T^{3} + 385665 p^{3} T^{4} - 234 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 673 T + 749898 T^{2} + 301087949 T^{3} + 749898 p^{3} T^{4} + 673 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 386 T + 549197 T^{2} + 181189028 T^{3} + 549197 p^{3} T^{4} + 386 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 16 T + 828849 T^{2} - 56595352 T^{3} + 828849 p^{3} T^{4} - 16 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 892 T + 466231 T^{2} + 343918592 T^{3} + 466231 p^{3} T^{4} + 892 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 1263 T + 1118004 T^{2} + 741292675 T^{3} + 1118004 p^{3} T^{4} + 1263 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 1815 T + 2216940 T^{2} + 2103744111 T^{3} + 2216940 p^{3} T^{4} + 1815 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 1800 T + 2584479 T^{2} - 2568723640 T^{3} + 2584479 p^{3} T^{4} - 1800 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 840 T + 1820643 T^{2} + 757404304 T^{3} + 1820643 p^{3} T^{4} + 840 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−8.146649469004933684224130183640, −7.68601623587707586930118913229, −7.51380784336560094511164234462, −7.42182523151609588089684203761, −7.07347963563550503502391400363, −6.93942687212484658800974692089, −6.67595106048498905148214056269, −6.11327339777450725188285239814, −5.93411563624558555385146452136, −5.91532471273555059391413711046, −5.16130447365789792495573832442, −4.99404661194005712066647994286, −4.93095394823730203375550937906, −4.43220851127969011432205508804, −4.36496597463423691042810972180, −4.12449522903152772273199369256, −3.53117768187808471979404951777, −3.32144502962487090008675734252, −3.20989566167867321409531826901, −2.57351058249551052627605258341, −2.49957748022951357093666982955, −2.05920507445793014945921678138, −1.43910009301941113965131973890, −1.14945861957921636046542169205, −1.05595781890755154060583478068, 0, 0, 0,
1.05595781890755154060583478068, 1.14945861957921636046542169205, 1.43910009301941113965131973890, 2.05920507445793014945921678138, 2.49957748022951357093666982955, 2.57351058249551052627605258341, 3.20989566167867321409531826901, 3.32144502962487090008675734252, 3.53117768187808471979404951777, 4.12449522903152772273199369256, 4.36496597463423691042810972180, 4.43220851127969011432205508804, 4.93095394823730203375550937906, 4.99404661194005712066647994286, 5.16130447365789792495573832442, 5.91532471273555059391413711046, 5.93411563624558555385146452136, 6.11327339777450725188285239814, 6.67595106048498905148214056269, 6.93942687212484658800974692089, 7.07347963563550503502391400363, 7.42182523151609588089684203761, 7.51380784336560094511164234462, 7.68601623587707586930118913229, 8.146649469004933684224130183640
