| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8201175767\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8201175767\) |
| \(L(\frac{5}{2})\) |
|
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
−8.554788506479603047017022812725, −8.016135375957208348443834637786, −7.64255872955725365873049718318, −7.60632838745308315320036375727, −7.38285853119257440531507471232, −7.17661723602044189991631373272, −6.77869701818573832443564971794, −6.37605361929033029909593553927, −6.12268297336176017096975554656, −6.00246619359919048314938982397, −5.21294015324364247135149799056, −5.11395257140662274878112368258, −5.06310329970279538011017737515, −4.38521978728473486030291783965, −4.22057694727016287778711216504, −4.12463576321131943517577708272, −3.36987185654270255202751829043, −3.21495782763885518988505697329, −3.01011863823493277133803274948, −2.41652052430402304832121621923, −2.01104205526282171519443229088, −1.98628157297843355747204002540, −0.810308513993315584394279169985, −0.71073809864372135639065978957, −0.21393516683669195349907019643,
0.21393516683669195349907019643, 0.71073809864372135639065978957, 0.810308513993315584394279169985, 1.98628157297843355747204002540, 2.01104205526282171519443229088, 2.41652052430402304832121621923, 3.01011863823493277133803274948, 3.21495782763885518988505697329, 3.36987185654270255202751829043, 4.12463576321131943517577708272, 4.22057694727016287778711216504, 4.38521978728473486030291783965, 5.06310329970279538011017737515, 5.11395257140662274878112368258, 5.21294015324364247135149799056, 6.00246619359919048314938982397, 6.12268297336176017096975554656, 6.37605361929033029909593553927, 6.77869701818573832443564971794, 7.17661723602044189991631373272, 7.38285853119257440531507471232, 7.60632838745308315320036375727, 7.64255872955725365873049718318, 8.016135375957208348443834637786, 8.554788506479603047017022812725
