| L(s) = 1 | + 24·2-s − 252·3-s + 2.04e3·4-s − 4.83e3·5-s − 6.04e3·6-s + 1.33e5·8-s + 1.77e5·9-s − 1.15e5·10-s − 5.34e5·11-s − 5.16e5·12-s − 1.15e6·13-s + 1.21e6·15-s + 3.20e6·16-s + 6.90e6·17-s + 4.25e6·18-s − 1.06e7·19-s − 9.89e6·20-s − 1.28e7·22-s − 1.86e7·23-s − 3.36e7·24-s + 4.88e7·25-s − 2.77e7·26-s − 1.17e8·27-s + 2.56e8·29-s + 2.92e7·30-s + 5.28e7·31-s + 2.73e8·32-s + ⋯ |
| L(s) = 1 | + 0.530·2-s − 0.598·3-s + 4-s − 0.691·5-s − 0.317·6-s + 1.44·8-s + 9-s − 0.366·10-s − 1.00·11-s − 0.598·12-s − 0.863·13-s + 0.413·15-s + 0.764·16-s + 1.17·17-s + 0.530·18-s − 0.987·19-s − 0.691·20-s − 0.530·22-s − 0.603·23-s − 0.863·24-s + 25-s − 0.457·26-s − 1.58·27-s + 2.32·29-s + 0.219·30-s + 0.331·31-s + 1.44·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(3.224242470\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.224242470\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 p^{3} T - 23 p^{6} T^{2} - 3 p^{14} T^{3} + p^{22} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 28 p^{2} T - 1403 p^{4} T^{2} + 28 p^{13} T^{3} + p^{22} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 966 p T - 1019969 p^{2} T^{2} + 966 p^{12} T^{3} + p^{22} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 534612 T + 498319933 T^{2} + 534612 p^{11} T^{3} + p^{22} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 577738 T + p^{11} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 6905934 T + 13420028104723 T^{2} - 6905934 p^{11} T^{3} + p^{22} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 10661420 T - 2824382481819 T^{2} + 10661420 p^{11} T^{3} + p^{22} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 18643272 T - 605238167047943 T^{2} + 18643272 p^{11} T^{3} + p^{22} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 128406630 T + p^{11} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 52843168 T - 22616076492128607 T^{2} - 52843168 p^{11} T^{3} + p^{22} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 182213314 T - 144715929980597817 T^{2} - 182213314 p^{11} T^{3} + p^{22} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 308120442 T + p^{11} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 17125708 T + p^{11} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 2687348496 T + 4749682723869449713 T^{2} + 2687348496 p^{11} T^{3} + p^{22} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 1596055698 T - 6721642138253924393 T^{2} - 1596055698 p^{11} T^{3} + p^{22} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 5189203740 T - 3228052989507855059 T^{2} - 5189203740 p^{11} T^{3} + p^{22} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 6956478662 T + 4878677763425471583 T^{2} + 6956478662 p^{11} T^{3} + p^{22} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 15481826884 T + \)\(11\!\cdots\!73\)\( T^{2} - 15481826884 p^{11} T^{3} + p^{22} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 9791485272 T + p^{11} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 1463791322 T - \)\(31\!\cdots\!93\)\( T^{2} + 1463791322 p^{11} T^{3} + p^{22} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 38116845680 T + \)\(70\!\cdots\!21\)\( T^{2} + 38116845680 p^{11} T^{3} + p^{22} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 29335099668 T + p^{11} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 24992917110 T - \)\(21\!\cdots\!89\)\( T^{2} - 24992917110 p^{11} T^{3} + p^{22} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 75013568546 T + p^{11} T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.13349775019279273151327031352, −12.99240613998643564270261948849, −12.25895772527501664081179199184, −11.84437114243267470670993820132, −11.26720874824352094659607559322, −10.55105311207253810108195228541, −10.23806418134221626567506843141, −9.720625330119521027864389416988, −8.236403130062008967508078268070, −7.940152678705052870808684871887, −7.24231622781944076694016136289, −6.75751022600719580827887321387, −6.01786870573064180676889169166, −5.05470568205721414125637847223, −4.63671004275728892678313703856, −3.96355925860512980909503467941, −2.89273903337605532608445643363, −2.23785162338011025142793011613, −1.31534622720672426529265478590, −0.51578049789593408582365466727,
0.51578049789593408582365466727, 1.31534622720672426529265478590, 2.23785162338011025142793011613, 2.89273903337605532608445643363, 3.96355925860512980909503467941, 4.63671004275728892678313703856, 5.05470568205721414125637847223, 6.01786870573064180676889169166, 6.75751022600719580827887321387, 7.24231622781944076694016136289, 7.940152678705052870808684871887, 8.236403130062008967508078268070, 9.720625330119521027864389416988, 10.23806418134221626567506843141, 10.55105311207253810108195228541, 11.26720874824352094659607559322, 11.84437114243267470670993820132, 12.25895772527501664081179199184, 12.99240613998643564270261948849, 13.13349775019279273151327031352
