| 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
| \(L(6)\) |
\(\approx\) |
\(10.82072818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.82072818\) |
| \(L(\frac{13}{2})\) |
|
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
−13.63127373498507838950278919510, −13.11652284482993564658038988533, −12.46723612742482659933225337368, −11.89659023202240636566138197641, −10.86739919228157851379995069197, −10.71780581200859915008162222552, −10.11319766071529462204154800299, −9.464609138985420190652111133750, −8.425552420259511262657977813724, −8.149083239662200641081063194734, −7.02550027721407422630161078079, −6.90012787818593900548102645527, −6.07292621766043728724281557183, −5.10083608107666506593453654550, −4.58666803203367537256189073990, −3.76003447954918115558626209742, −2.63266405292909951432822305995, −2.44655220540359277797802574558, −1.43981670022813946246416561018, −0.900005096472230294746675404874,
0.900005096472230294746675404874, 1.43981670022813946246416561018, 2.44655220540359277797802574558, 2.63266405292909951432822305995, 3.76003447954918115558626209742, 4.58666803203367537256189073990, 5.10083608107666506593453654550, 6.07292621766043728724281557183, 6.90012787818593900548102645527, 7.02550027721407422630161078079, 8.149083239662200641081063194734, 8.425552420259511262657977813724, 9.464609138985420190652111133750, 10.11319766071529462204154800299, 10.71780581200859915008162222552, 10.86739919228157851379995069197, 11.89659023202240636566138197641, 12.46723612742482659933225337368, 13.11652284482993564658038988533, 13.63127373498507838950278919510
