| L(s) = 1 | − 16·2-s + 192·4-s + 142·5-s − 470·7-s − 2.04e3·8-s − 2.27e3·10-s − 2.63e3·11-s + 7.52e3·14-s + 2.04e4·16-s + 2.72e4·20-s + 4.20e4·22-s + 1.56e4·25-s − 9.02e4·28-s − 2.30e3·29-s + 5.46e4·31-s − 1.96e5·32-s − 6.67e4·35-s − 2.90e5·40-s − 5.04e5·44-s + 1.17e5·49-s − 2.50e5·50-s + 3.84e4·53-s − 3.73e5·55-s + 9.62e5·56-s + 3.68e4·58-s + 2.06e5·59-s − 8.74e5·62-s + ⋯ |
| L(s) = 1 | − 2·2-s + 3·4-s + 1.13·5-s − 1.37·7-s − 4·8-s − 2.27·10-s − 1.97·11-s + 2.74·14-s + 5·16-s + 3.40·20-s + 3.95·22-s + 25-s − 4.11·28-s − 0.0943·29-s + 1.83·31-s − 6·32-s − 1.55·35-s − 4.54·40-s − 5.92·44-s + 49-s − 2·50-s + 0.258·53-s − 2.24·55-s + 5.48·56-s + 0.188·58-s + 1.00·59-s − 3.67·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.120821514\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.120821514\) |
| \(L(4)\) |
|
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 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 142 T + 4539 T^{2} - 142 p^{6} T^{3} + p^{12} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 470 T + 103251 T^{2} + 470 p^{6} T^{3} + p^{12} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2630 T + 5145339 T^{2} + 2630 p^{6} T^{3} + p^{12} T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 1150 T + p^{6} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 54682 T + 2102617443 T^{2} - 54682 p^{6} T^{3} + p^{12} T^{4} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 38446 T - 20686266213 T^{2} - 38446 p^{6} T^{3} + p^{12} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 103430 T + p^{6} T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 674350 T + 303413696211 T^{2} - 674350 p^{6} T^{3} + p^{12} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 890822 T + p^{6} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 363274 T - 194972374293 T^{2} - 363274 p^{6} T^{3} + p^{12} T^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1495870 T + 1404655051971 T^{2} - 1495870 p^{6} T^{3} + p^{12} T^{4} \) |
| 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
−11.16606678514887364645101849772, −10.61081677252487433962014429607, −10.31327638894459821560679942309, −10.03405354799611633730972979663, −9.573170497495282916292493299707, −9.196744785296293781649039552168, −8.603534990534067637483810395972, −8.065737933066201679163213211695, −7.67047529363358717176961433574, −7.01583857828002489010421846931, −6.39279220170267721410550392785, −6.24815762094365991933600108501, −5.51278163062920859653123914074, −4.98261010140904316582905518617, −3.45098228479435734384953701284, −2.97047466431638725589750880387, −2.31018607439892561349859063462, −2.05898862350332563214654642917, −0.71529742234599809146725436633, −0.58897739277851823822468445471,
0.58897739277851823822468445471, 0.71529742234599809146725436633, 2.05898862350332563214654642917, 2.31018607439892561349859063462, 2.97047466431638725589750880387, 3.45098228479435734384953701284, 4.98261010140904316582905518617, 5.51278163062920859653123914074, 6.24815762094365991933600108501, 6.39279220170267721410550392785, 7.01583857828002489010421846931, 7.67047529363358717176961433574, 8.065737933066201679163213211695, 8.603534990534067637483810395972, 9.196744785296293781649039552168, 9.573170497495282916292493299707, 10.03405354799611633730972979663, 10.31327638894459821560679942309, 10.61081677252487433962014429607, 11.16606678514887364645101849772
