| L(s) = 1 | − 300·5-s − 243·9-s − 6.78e3·13-s + 1.03e4·17-s + 3.62e4·25-s − 6.42e4·29-s + 1.52e5·37-s − 1.40e5·41-s + 7.29e4·45-s + 1.29e5·49-s − 1.33e5·53-s + 5.14e5·61-s + 2.03e6·65-s + 4.86e5·73-s + 5.90e4·81-s − 3.10e6·85-s − 1.37e6·89-s − 1.88e6·97-s + 5.19e5·101-s + 2.04e6·109-s − 2.62e6·113-s + 1.64e6·117-s + 1.36e6·121-s + 5.62e5·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 2.39·5-s − 1/3·9-s − 3.08·13-s + 2.10·17-s + 2.31·25-s − 2.63·29-s + 3.00·37-s − 2.03·41-s + 4/5·45-s + 1.09·49-s − 0.899·53-s + 2.26·61-s + 7.41·65-s + 1.25·73-s + 1/9·81-s − 5.05·85-s − 1.94·89-s − 2.06·97-s + 0.504·101-s + 1.58·109-s − 1.82·113-s + 1.02·117-s + 0.770·121-s + 0.287·125-s + 6.32·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(9.563712700\times10^{-6}\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.563712700\times10^{-6}\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p^{5} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 6 p^{2} T + p^{6} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 129266 T^{2} + p^{12} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 1365410 T^{2} + p^{12} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 3394 T + p^{6} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 5178 T + p^{6} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 47709890 T^{2} + p^{12} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 280146530 T^{2} + p^{12} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 32142 T + p^{6} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 711526610 T^{2} + p^{12} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 76150 T + p^{6} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 70038 T + p^{6} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 2451652130 T^{2} + p^{12} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 1425021694 T^{2} + p^{12} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 66942 T + p^{6} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 68178858910 T^{2} + p^{12} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 257014 T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 77748332930 T^{2} + p^{12} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 138474492194 T^{2} + p^{12} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 243442 T + p^{6} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 260585085842 T^{2} + p^{12} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 415389237694 T^{2} + p^{12} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 686766 T + p^{6} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 942686 T + p^{6} T^{2} )^{2} \) |
| 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
−12.08644960372799131905528015474, −11.44075755328147395947945165349, −10.96287907639051323708870438404, −9.996998108111873356103877594881, −9.753768232606397729234863922061, −9.420794360761443201177363175831, −8.362207430228901083335386668424, −7.989416789333918966356810791783, −7.56166096547342427925018623594, −7.40586034193957804923174789471, −6.88596438807533889093712544667, −5.69652810973822116883021461183, −5.28927207194301954959714402871, −4.66241917898031283860126566046, −3.99433832887367642840355147350, −3.57409448465980891543459708572, −2.86554668320248084744490069978, −2.18095745078240950625752512643, −0.911667414241835449930565762604, −0.00108339349063662211504236978,
0.00108339349063662211504236978, 0.911667414241835449930565762604, 2.18095745078240950625752512643, 2.86554668320248084744490069978, 3.57409448465980891543459708572, 3.99433832887367642840355147350, 4.66241917898031283860126566046, 5.28927207194301954959714402871, 5.69652810973822116883021461183, 6.88596438807533889093712544667, 7.40586034193957804923174789471, 7.56166096547342427925018623594, 7.989416789333918966356810791783, 8.362207430228901083335386668424, 9.420794360761443201177363175831, 9.753768232606397729234863922061, 9.996998108111873356103877594881, 10.96287907639051323708870438404, 11.44075755328147395947945165349, 12.08644960372799131905528015474
