| L(s) = 1 | + 2·3-s − 2·5-s − 2·7-s + 3·9-s + 4·11-s + 2·13-s − 4·15-s + 6·19-s − 4·21-s − 4·23-s − 2·25-s + 4·27-s − 2·31-s + 8·33-s + 4·35-s + 8·37-s + 4·39-s − 2·41-s + 4·43-s − 6·45-s + 4·47-s − 6·49-s − 8·53-s − 8·55-s + 12·57-s + 8·59-s + 16·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s − 0.755·7-s + 9-s + 1.20·11-s + 0.554·13-s − 1.03·15-s + 1.37·19-s − 0.872·21-s − 0.834·23-s − 2/5·25-s + 0.769·27-s − 0.359·31-s + 1.39·33-s + 0.676·35-s + 1.31·37-s + 0.640·39-s − 0.312·41-s + 0.609·43-s − 0.894·45-s + 0.583·47-s − 6/7·49-s − 1.09·53-s − 1.07·55-s + 1.58·57-s + 1.04·59-s + 2.04·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.599593000\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.599593000\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 42 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 78 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_4$ | \( 1 - 8 T + 114 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_4$ | \( 1 - 16 T + 166 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 18 T + 210 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 122 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 24 T + 290 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 54 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 174 T^{2} + p^{2} T^{4} \) |
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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
−9.094980197561889327144857358578, −8.778632548233681391904241735905, −8.228043260112765692921998281977, −8.068248438097710411643265375831, −7.55128139756572079991675566549, −7.46204735309805465217878330782, −6.69557794232199301015024689804, −6.66292509619315579324153902284, −6.11700049276601143871233485110, −5.67634836837628061554418660925, −5.13492977163016862431807701878, −4.58573884256097150166761174127, −3.97887609829831420641421868557, −3.89606966967022440095906535683, −3.34439181217073549366725737017, −3.26328767652677035002458517137, −2.37459007471475084845597070853, −2.03950127016774013614693680851, −1.19062599882514156612810309764, −0.66088564703388225859422368559,
0.66088564703388225859422368559, 1.19062599882514156612810309764, 2.03950127016774013614693680851, 2.37459007471475084845597070853, 3.26328767652677035002458517137, 3.34439181217073549366725737017, 3.89606966967022440095906535683, 3.97887609829831420641421868557, 4.58573884256097150166761174127, 5.13492977163016862431807701878, 5.67634836837628061554418660925, 6.11700049276601143871233485110, 6.66292509619315579324153902284, 6.69557794232199301015024689804, 7.46204735309805465217878330782, 7.55128139756572079991675566549, 8.068248438097710411643265375831, 8.228043260112765692921998281977, 8.778632548233681391904241735905, 9.094980197561889327144857358578
