| L(s) = 1 | + 9·3-s + 11·5-s + 54·9-s + 19·11-s + 22·13-s + 99·15-s + 104·17-s + 202·19-s + 280·23-s − 34·25-s + 270·27-s − 73·29-s − 131·31-s + 171·33-s − 326·37-s + 198·39-s + 516·41-s − 36·43-s + 594·45-s − 126·47-s + 936·51-s − 385·53-s + 209·55-s + 1.81e3·57-s + 285·59-s + 34·61-s + 242·65-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.983·5-s + 2·9-s + 0.520·11-s + 0.469·13-s + 1.70·15-s + 1.48·17-s + 2.43·19-s + 2.53·23-s − 0.271·25-s + 1.92·27-s − 0.467·29-s − 0.758·31-s + 0.902·33-s − 1.44·37-s + 0.812·39-s + 1.96·41-s − 0.127·43-s + 1.96·45-s − 0.391·47-s + 2.56·51-s − 0.997·53-s + 0.512·55-s + 4.22·57-s + 0.628·59-s + 0.0713·61-s + 0.461·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(36.81500880\) |
| \(L(\frac12)\) |
\(\approx\) |
\(36.81500880\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $S_4\times C_2$ | \( 1 - 11 T + 31 p T^{2} - 2802 T^{3} + 31 p^{4} T^{4} - 11 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 19 T + 3369 T^{2} - 53854 T^{3} + 3369 p^{3} T^{4} - 19 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 22 T + 3096 T^{2} - 123644 T^{3} + 3096 p^{3} T^{4} - 22 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 104 T + 17299 T^{2} - 1017872 T^{3} + 17299 p^{3} T^{4} - 104 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 202 T + 13334 T^{2} - 545068 T^{3} + 13334 p^{3} T^{4} - 202 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 280 T + 43477 T^{2} - 5280784 T^{3} + 43477 p^{3} T^{4} - 280 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 73 T + 44143 T^{2} + 4531786 T^{3} + 44143 p^{3} T^{4} + 73 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 131 T + 41504 T^{2} + 3648635 T^{3} + 41504 p^{3} T^{4} + 131 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 326 T + 146712 T^{2} + 33007960 T^{3} + 146712 p^{3} T^{4} + 326 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 516 T + 211167 T^{2} - 56124328 T^{3} + 211167 p^{3} T^{4} - 516 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 36 T + 97410 T^{2} - 1479718 T^{3} + 97410 p^{3} T^{4} + 36 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 126 T + 161529 T^{2} + 37845548 T^{3} + 161529 p^{3} T^{4} + 126 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 385 T + 426499 T^{2} + 114813418 T^{3} + 426499 p^{3} T^{4} + 385 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 285 T + 565605 T^{2} - 115272798 T^{3} + 565605 p^{3} T^{4} - 285 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 34 T + 387547 T^{2} - 37674860 T^{3} + 387547 p^{3} T^{4} - 34 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 100 T + 755250 T^{2} - 32905634 T^{3} + 755250 p^{3} T^{4} - 100 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 34 T + 294609 T^{2} - 182711756 T^{3} + 294609 p^{3} T^{4} + 34 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 108 T + 188712 T^{2} + 325630402 T^{3} + 188712 p^{3} T^{4} - 108 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 2463 T + 3428340 T^{2} - 2921490875 T^{3} + 3428340 p^{3} T^{4} - 2463 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 115 T + 1228989 T^{2} + 243220678 T^{3} + 1228989 p^{3} T^{4} + 115 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 110 T + 387883 T^{2} + 192185284 T^{3} + 387883 p^{3} T^{4} - 110 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 2941 T + 5549515 T^{2} - 6235037870 T^{3} + 5549515 p^{3} T^{4} - 2941 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−7.73204691731937448992145081302, −7.43503885218563591551834849183, −7.20446852447837886263918285206, −7.05659928293501553883167221338, −6.44153208861240013995473055769, −6.31962950132854751186267309204, −6.30091673473803853746851965054, −5.59550012074656380189609105971, −5.35532190578459440178629268053, −5.28049721282421700354155723344, −5.07303604477210780946354667539, −4.66625675393868281850187475152, −4.30964654533802490617324426283, −3.69213941999899749650639429257, −3.66084924426145258091158332356, −3.46591386918302311397727857709, −3.13989103882122665967192624455, −2.79408338859058747538033365355, −2.72416162699649128734181271037, −1.94258543196174633309268858082, −1.85760923707588160451933435696, −1.63249503149926929044577535102, −1.01253358967009164274818646198, −0.818015891927076990246539509357, −0.67478570353209734172375947942,
0.67478570353209734172375947942, 0.818015891927076990246539509357, 1.01253358967009164274818646198, 1.63249503149926929044577535102, 1.85760923707588160451933435696, 1.94258543196174633309268858082, 2.72416162699649128734181271037, 2.79408338859058747538033365355, 3.13989103882122665967192624455, 3.46591386918302311397727857709, 3.66084924426145258091158332356, 3.69213941999899749650639429257, 4.30964654533802490617324426283, 4.66625675393868281850187475152, 5.07303604477210780946354667539, 5.28049721282421700354155723344, 5.35532190578459440178629268053, 5.59550012074656380189609105971, 6.30091673473803853746851965054, 6.31962950132854751186267309204, 6.44153208861240013995473055769, 7.05659928293501553883167221338, 7.20446852447837886263918285206, 7.43503885218563591551834849183, 7.73204691731937448992145081302
