| L(s) = 1 | + 2·3-s + 9-s + 4·11-s + 8·13-s − 16·23-s + 2·25-s − 4·27-s + 8·33-s + 8·37-s + 16·39-s + 6·49-s + 12·59-s + 8·61-s − 32·69-s − 16·71-s + 20·73-s + 4·75-s − 11·81-s + 12·83-s − 12·97-s + 4·99-s + 12·107-s + 8·109-s + 16·111-s + 8·117-s − 10·121-s + 127-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s + 1.20·11-s + 2.21·13-s − 3.33·23-s + 2/5·25-s − 0.769·27-s + 1.39·33-s + 1.31·37-s + 2.56·39-s + 6/7·49-s + 1.56·59-s + 1.02·61-s − 3.85·69-s − 1.89·71-s + 2.34·73-s + 0.461·75-s − 1.22·81-s + 1.31·83-s − 1.21·97-s + 0.402·99-s + 1.16·107-s + 0.766·109-s + 1.51·111-s + 0.739·117-s − 0.909·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.443905470\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.443905470\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−10.51790162432362088151821340791, −9.855412101229841779083687311730, −9.765423492759884298144469063424, −8.994596815513396479366892225998, −8.961369816676942957874767219788, −8.281710285438635550797649542370, −8.182631434351678903266495681889, −7.76900236477270828771675802500, −7.12465929549730367019309647976, −6.44171877092891243843215888464, −6.22670110820432856620595626044, −5.84569945370550628814190632473, −5.27042862733379677804268421892, −4.15801066384631971584738330440, −4.00130827318759063374247396910, −3.77015201184958142725796690470, −3.07660828285776428562006821508, −2.25943473043660764430411741962, −1.80494644618733160370409467942, −0.944281193568862877142394867146,
0.944281193568862877142394867146, 1.80494644618733160370409467942, 2.25943473043660764430411741962, 3.07660828285776428562006821508, 3.77015201184958142725796690470, 4.00130827318759063374247396910, 4.15801066384631971584738330440, 5.27042862733379677804268421892, 5.84569945370550628814190632473, 6.22670110820432856620595626044, 6.44171877092891243843215888464, 7.12465929549730367019309647976, 7.76900236477270828771675802500, 8.182631434351678903266495681889, 8.281710285438635550797649542370, 8.961369816676942957874767219788, 8.994596815513396479366892225998, 9.765423492759884298144469063424, 9.855412101229841779083687311730, 10.51790162432362088151821340791
