| L(s) = 1 | + 88·4-s + 3.76e3·16-s − 1.93e3·19-s + 3.83e3·25-s + 1.44e4·31-s − 3.35e4·49-s + 8.54e4·61-s + 7.04e4·64-s − 1.70e5·76-s − 3.98e5·79-s + 3.37e5·100-s + 6.77e3·109-s − 1.40e5·121-s + 1.27e6·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.38e6·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 11/4·4-s + 3.67·16-s − 1.23·19-s + 1.22·25-s + 2.69·31-s − 1.99·49-s + 2.94·61-s + 2.14·64-s − 3.38·76-s − 7.18·79-s + 3.37·100-s + 0.0546·109-s − 0.870·121-s + 7.41·124-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 3.72·169-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(6.909729436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.909729436\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - 766 p T^{2} + p^{10} T^{4} \) |
| good | 2 | $C_2^2$ | \( ( 1 - 11 p^{2} T^{2} + p^{10} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 2398 p T^{2} + p^{10} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 70102 T^{2} + p^{10} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 692186 T^{2} + p^{10} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 57134 T^{2} + p^{10} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 484 T + p^{5} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 14654 p^{2} T^{2} + p^{10} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 10530298 T^{2} + p^{10} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 3608 T + p^{5} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 83802314 T^{2} + p^{10} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 109744402 T^{2} + p^{10} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 135962486 T^{2} + p^{10} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 367610894 T^{2} + p^{10} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 813621206 T^{2} + p^{10} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 1399356598 T^{2} + p^{10} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 21362 T + p^{5} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 1504963814 T^{2} + p^{10} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 2510746702 T^{2} + p^{10} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 4138885586 T^{2} + p^{10} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 99616 T + p^{5} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 4615601606 T^{2} + p^{10} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 3347081102 T^{2} + p^{10} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 13052162114 T^{2} + p^{10} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.83224676582407332869034412173, −10.66341554480011279125419100584, −10.06257577487544824513806148999, −9.994062621061679483891007575304, −9.875688674686054121279194008582, −9.038678790073071448155169786653, −8.549920576038356203551742193463, −8.511137356933303855154340509177, −8.038277402242260252965844974223, −7.70530314059212406696192150973, −7.02392205339127744933052634002, −6.94863621667277357273183464548, −6.82059201830496534405408186100, −6.24346348423030527136328636253, −6.12352382054055081290956472604, −5.60942258597415453355976194803, −5.06591553333735791787709444121, −4.32768687810259873268172939927, −4.23536654593675827458714478007, −3.04446326279113337484234757229, −2.93966998746581877814460196130, −2.57913545993673695142714321798, −1.82450157580525235902138059426, −1.51537898602479532329673229826, −0.61890446605563412164653789820,
0.61890446605563412164653789820, 1.51537898602479532329673229826, 1.82450157580525235902138059426, 2.57913545993673695142714321798, 2.93966998746581877814460196130, 3.04446326279113337484234757229, 4.23536654593675827458714478007, 4.32768687810259873268172939927, 5.06591553333735791787709444121, 5.60942258597415453355976194803, 6.12352382054055081290956472604, 6.24346348423030527136328636253, 6.82059201830496534405408186100, 6.94863621667277357273183464548, 7.02392205339127744933052634002, 7.70530314059212406696192150973, 8.038277402242260252965844974223, 8.511137356933303855154340509177, 8.549920576038356203551742193463, 9.038678790073071448155169786653, 9.875688674686054121279194008582, 9.994062621061679483891007575304, 10.06257577487544824513806148999, 10.66341554480011279125419100584, 10.83224676582407332869034412173
