| L(s) = 1 | − 516·5-s + 3.82e4·13-s + 1.17e5·17-s − 5.81e5·25-s − 1.68e6·29-s + 5.09e6·37-s + 8.64e6·41-s + 6.53e5·49-s − 2.38e6·53-s + 1.68e7·61-s − 1.97e7·65-s + 2.54e7·73-s − 6.05e7·85-s + 3.36e7·89-s + 2.41e8·97-s − 2.32e7·101-s + 3.11e7·109-s + 9.28e7·113-s − 1.07e8·121-s + 5.35e8·125-s + 127-s + 131-s + 137-s + 139-s + 8.69e8·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 0.825·5-s + 1.34·13-s + 1.40·17-s − 1.48·25-s − 2.38·29-s + 2.71·37-s + 3.06·41-s + 0.113·49-s − 0.302·53-s + 1.21·61-s − 1.10·65-s + 0.896·73-s − 1.16·85-s + 0.535·89-s + 2.73·97-s − 0.223·101-s + 0.220·109-s + 0.569·113-s − 0.500·121-s + 2.19·125-s + 1.96·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.954105480\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.954105480\) |
| \(L(5)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + 258 T + p^{8} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 13346 p^{2} T^{2} + p^{16} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 107392510 T^{2} + p^{16} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 19138 T + p^{8} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 58686 T + p^{8} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 10688638850 T^{2} + p^{16} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 83658891650 T^{2} + p^{16} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 842178 T + p^{8} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 600755547650 T^{2} + p^{16} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2548610 T + p^{8} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 4324158 T + p^{8} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 19226964507650 T^{2} + p^{16} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 4696760080126 T^{2} + p^{16} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 1192194 T + p^{8} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 293546462999810 T^{2} + p^{16} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8414786 T + p^{8} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 508514414548610 T^{2} + p^{16} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 338146626840194 T^{2} + p^{16} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 12735874 T + p^{8} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 2994695631333122 T^{2} + p^{16} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 2396699180600446 T^{2} + p^{16} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 16802814 T + p^{8} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 120994882 T + p^{8} T^{2} )^{2} \) |
| 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
−11.51500545628896064363547167101, −11.34924397919114580665287674027, −11.11110379246237527525442196911, −10.30718662352294283140860814112, −9.582714756451410324195525793358, −9.418934951514100026924217239462, −8.681326783072262849125286540161, −7.894496460259830982251139442673, −7.69754505337975148311002367702, −7.39548917643654196673661552758, −6.09725657392604548044663549738, −6.06324258592219373302082506514, −5.41929220642023390797973897952, −4.42326317764325366211256039943, −3.74363140252888085108812247937, −3.69851166121830373604393325739, −2.63891350811472962659918968214, −1.87641713017570057694015655428, −0.964337835439965625524582881827, −0.54726458824591683671650108890,
0.54726458824591683671650108890, 0.964337835439965625524582881827, 1.87641713017570057694015655428, 2.63891350811472962659918968214, 3.69851166121830373604393325739, 3.74363140252888085108812247937, 4.42326317764325366211256039943, 5.41929220642023390797973897952, 6.06324258592219373302082506514, 6.09725657392604548044663549738, 7.39548917643654196673661552758, 7.69754505337975148311002367702, 7.894496460259830982251139442673, 8.681326783072262849125286540161, 9.418934951514100026924217239462, 9.582714756451410324195525793358, 10.30718662352294283140860814112, 11.11110379246237527525442196911, 11.34924397919114580665287674027, 11.51500545628896064363547167101
