| L(s) = 1 | + 78·2-s + 2.04e3·4-s − 5.37e3·5-s + 2.77e4·7-s + 4.68e3·8-s − 4.18e5·10-s + 6.37e5·11-s − 7.66e5·13-s + 2.16e6·14-s + 3.65e5·16-s − 6.16e6·17-s − 3.90e7·19-s − 1.09e7·20-s + 4.97e7·22-s + 1.53e7·23-s + 4.88e7·25-s − 5.97e7·26-s + 5.68e7·28-s + 1.07e7·29-s + 5.09e7·31-s + 9.58e6·32-s − 4.81e8·34-s − 1.49e8·35-s + 1.32e9·37-s − 3.04e9·38-s − 2.51e7·40-s + 8.98e8·41-s + ⋯ |
| L(s) = 1 | + 1.72·2-s + 4-s − 0.768·5-s + 0.624·7-s + 0.0504·8-s − 1.32·10-s + 1.19·11-s − 0.572·13-s + 1.07·14-s + 0.0870·16-s − 1.05·17-s − 3.61·19-s − 0.768·20-s + 2.05·22-s + 0.496·23-s + 25-s − 0.986·26-s + 0.624·28-s + 0.0973·29-s + 0.319·31-s + 0.0504·32-s − 1.81·34-s − 0.479·35-s + 3.15·37-s − 6.23·38-s − 0.0388·40-s + 1.21·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(4.001956212\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.001956212\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 39 p T + 1009 p^{2} T^{2} - 39 p^{12} T^{3} + p^{22} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 1074 p T - 799649 p^{2} T^{2} + 1074 p^{12} T^{3} + p^{22} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 27760 T - 1206709143 T^{2} - 27760 p^{11} T^{3} + p^{22} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 637836 T + 121523092285 T^{2} - 637836 p^{11} T^{3} + p^{22} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 766214 T - 1205076500241 T^{2} + 766214 p^{11} T^{3} + p^{22} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 3084354 T + p^{11} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 1026916 p T + p^{11} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 15312360 T - 718341389144327 T^{2} - 15312360 p^{11} T^{3} + p^{22} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10751262 T - 12084920131113185 T^{2} - 10751262 p^{11} T^{3} + p^{22} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 50937400 T - 22813858177644831 T^{2} - 50937400 p^{11} T^{3} + p^{22} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 664740830 T + p^{11} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 898833450 T + 257572539122654059 T^{2} - 898833450 p^{11} T^{3} + p^{22} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 957947188 T - 11630924474115363 T^{2} - 957947188 p^{11} T^{3} + p^{22} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 1555741344 T - 51828085653085967 T^{2} + 1555741344 p^{11} T^{3} + p^{22} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 3792417030 T + p^{11} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 555306924 T - 29847522664895500883 T^{2} - 555306924 p^{11} T^{3} + p^{22} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 4950420998 T - 19007249553996522657 T^{2} + 4950420998 p^{11} T^{3} + p^{22} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 5292399284 T - 94120642723684304427 T^{2} + 5292399284 p^{11} T^{3} + p^{22} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 14831086248 T + p^{11} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 13971005210 T + p^{11} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 3720542360 T - \)\(73\!\cdots\!79\)\( T^{2} + 3720542360 p^{11} T^{3} + p^{22} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 8768454036 T - \)\(12\!\cdots\!71\)\( T^{2} - 8768454036 p^{11} T^{3} + p^{22} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 25472769174 T + p^{11} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 39092494846 T - \)\(56\!\cdots\!37\)\( T^{2} - 39092494846 p^{11} T^{3} + p^{22} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.11215664602385107194272977196, −12.24899929254597359429373322056, −11.25156076254746166998604602883, −10.97022500519927352512357603502, −10.69896395328426401811706637753, −9.310115664869278777501182440321, −9.214647357935758646938249326612, −8.124242050128200712634844298775, −8.062697900965566829609063336082, −6.95191229657837817845357266693, −6.29843651556212874023769557961, −6.15498479276903752156961097800, −4.79888170807912448746529478546, −4.68955135337533552514672721750, −4.01368461311301499920948955531, −3.99655996671882731014820617686, −2.69818216519492721293403845827, −2.26463857484741169562861023511, −1.27372012588068908729071916165, −0.38116024685868552508610705155,
0.38116024685868552508610705155, 1.27372012588068908729071916165, 2.26463857484741169562861023511, 2.69818216519492721293403845827, 3.99655996671882731014820617686, 4.01368461311301499920948955531, 4.68955135337533552514672721750, 4.79888170807912448746529478546, 6.15498479276903752156961097800, 6.29843651556212874023769557961, 6.95191229657837817845357266693, 8.062697900965566829609063336082, 8.124242050128200712634844298775, 9.214647357935758646938249326612, 9.310115664869278777501182440321, 10.69896395328426401811706637753, 10.97022500519927352512357603502, 11.25156076254746166998604602883, 12.24899929254597359429373322056, 13.11215664602385107194272977196
