| L(s) = 1 | − 8·3-s + 5·4-s + 4·9-s − 40·12-s + 9·16-s − 88·23-s + 2·25-s + 200·27-s + 200·29-s + 20·36-s + 152·43-s − 72·48-s + 100·49-s + 200·61-s − 35·64-s + 704·69-s − 16·75-s − 790·81-s − 1.60e3·87-s − 440·92-s + 10·100-s − 88·101-s + 392·103-s − 808·107-s + 1.00e3·108-s + 1.00e3·116-s − 380·121-s + ⋯ |
| L(s) = 1 | − 8/3·3-s + 5/4·4-s + 4/9·9-s − 3.33·12-s + 9/16·16-s − 3.82·23-s + 2/25·25-s + 7.40·27-s + 6.89·29-s + 5/9·36-s + 3.53·43-s − 3/2·48-s + 2.04·49-s + 3.27·61-s − 0.546·64-s + 10.2·69-s − 0.213·75-s − 9.75·81-s − 18.3·87-s − 4.78·92-s + 1/10·100-s − 0.871·101-s + 3.80·103-s − 7.55·107-s + 9.25·108-s + 8.62·116-s − 3.14·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8602777481\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8602777481\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2^2$ | \( 1 - 5 T^{2} + p^{4} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 22 p T^{2} + p^{4} T^{4} \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{4} \) |
| 7 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 190 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 670 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 22 T + p^{2} T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 - 50 T + p^{2} T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 626 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 2686 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 2594 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 38 T + p^{2} T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 3218 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 670 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 50 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 3170 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 8782 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 1870 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 9986 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 94 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 11806 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 18766 T^{2} + p^{4} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.251397591281781460493180818068, −8.195445708602681169234623723345, −8.117693180172915269209987962240, −7.927621139818456336732735788935, −7.13378777733440119769788473865, −7.04046654083476843070978501448, −6.68241965366075778397661332162, −6.44984778137358505462576340964, −6.22495625800946684211645853332, −6.13262530901395824638212402287, −5.85596712064714944548500221569, −5.83546485576137689274030470959, −5.15838005216526800208584501286, −5.03977617029561532733162915150, −5.03421511841423492986400539067, −4.28349294228164863255068900727, −4.07291603638776239361131937319, −3.86768873748623462549984520448, −2.80728398016586938075468380932, −2.73704646093617345982198565550, −2.52559050964128914675671508840, −2.45332414520980113995666648242, −1.14990210814224420331520649950, −0.854582854444805411258489561890, −0.39734362206731051336079907847,
0.39734362206731051336079907847, 0.854582854444805411258489561890, 1.14990210814224420331520649950, 2.45332414520980113995666648242, 2.52559050964128914675671508840, 2.73704646093617345982198565550, 2.80728398016586938075468380932, 3.86768873748623462549984520448, 4.07291603638776239361131937319, 4.28349294228164863255068900727, 5.03421511841423492986400539067, 5.03977617029561532733162915150, 5.15838005216526800208584501286, 5.83546485576137689274030470959, 5.85596712064714944548500221569, 6.13262530901395824638212402287, 6.22495625800946684211645853332, 6.44984778137358505462576340964, 6.68241965366075778397661332162, 7.04046654083476843070978501448, 7.13378777733440119769788473865, 7.927621139818456336732735788935, 8.117693180172915269209987962240, 8.195445708602681169234623723345, 8.251397591281781460493180818068
